Mullah Abdul Rauf Khadim looked like any ubiquitous insurgent commander in southern Afghanistan. He had a sunbaked complexion, serried black beard, charcoal eyes, and the usual accessory – an AK-47 slung over his shoulder.

But there was something distinctive about him, which alarmed American officials. He had recently defected from the Taliban and joined Islamic State (known as both IS and ISIS), creating concern that the militant extremist group was expanding its footprint in South Asia.

So on Feb. 9, a US aircraft locked onto the vehicle he was traveling in near the village of Sadat in Helmand Province. It fired a missile, killing Mr. Khadim and five of his companions.

“The Islamic State is increasingly active in the region,” says a senior American military official in Kabul, Afghanistan, though cautioning not to inflate their size or significance – at least not yet. “Some locals appear to be attracted to their battlefield success in Iraq. And everyone loves a winner.”

A year ago, the prospect that IS might emerge in South Asia, the birthplace of Al Qaeda, seemed preposterous. True, IS operatives and their Sunni allies had pushed into western Iraq, seizing the cities of Fallujah and Ramadi, and moved rapidly across other parts of the country.

But they had yet to establish much of a presence elsewhere in the restive Islamic belt, including Pakistan and Afghanistan. These, after all, were strongholds of Al Qaeda, traditionally a fierce IS competitor, and the Taliban. But since last fall, IS has been slowly and methodically forging ties with militant groups in these two countries as well as other places around the globe.

“The initial ISIS reports began as rumors,” notes an Afghan defense official. “But not anymore.”

IS efforts to gain a foothold in South Asia and other regions highlight a disturbing trend. Islamic extremism is rising in key areas of the world.

Driven by a lack of stable governments and the movement of trained and ideologically committed recruits from battlefields in Iraq and Syria, extremist groups – such as IS and Al Qaeda – are spreading their reach into new areas of Africa, the Middle East, and South Asia. They are becoming more sophisticated in their communications, more lethal in their tactics, and more adept at fundraising.

But while some groups may be working together, creating the specter of a worldwide movement, deep fissures persist among the groups that will likely prevent them from becoming a global network.

Many Islamic fighters disagree about how much, if at all, to target Western countries and their citizens. Others disagree about the size and global nature of their desired emirate, the legitimacy of attacking Shiite Muslims, and the morality of killing civilians. In some countries, such as Syria, extremists have even engaged in intense battles with each other, widening already significant splits.

For all the strengths of today’s Islamic extremists, most are not committed to – or even capable of – conducting sophisticated attacks in the West. What’s more, polls show there is little popular support for most groups. Over the long run, their lack of local support and legitimacy may well undermine any fleeting gains – and the threat they pose to the West.

• • •

Competing networks

The narrow valleys and swelling rivers of the Hindu Kush mountains, along the Afghan-Pakistani border, make the terrain inhospitable. But it was here, nearly three decades ago, that Osama bin Laden and Ayman al-Zawahiri started Al Qaeda in the waning years of the war against the Soviets.

Bearded tribesmen clog the streets of many of the border towns, clad in their dusty sandals and shalwar kameez, the loose-fitting trousers and long, baggy shirts worn by locals. Most of Al Qaeda’s surviving leaders still remain in the area, despite the attempts by IS to recruit here.

Today, the terrorist landscape centers around these two broad movements: Al Qaeda and IS.

Al Qaeda is led by Mr. Zawahiri, the fiery Egyptian who took over when Mr. bin Laden was killed by US Navy SEALs in 2011. Al Qaeda’s goal remains establishing a loose Islamic caliphate that extends from Africa through the Middle East, South Asia, and parts of the Pacific.

Al Qaeda’s primary strategy from its base here is to work with its affiliates – such as Al Shabab in Somalia, Al Qaeda in the Arabian Peninsula in Yemen, Al Qaeda in the Islamic Maghreb in North Africa, and Jabhat al-Nusra in Syria – to overthrow local regimes. Zawahiri and his colleagues seek to replace these governments with ones that implement an extreme interpretation of Islamic law, or sharia.

“Al Qaeda leaders continue to encourage their affiliates to create states,” says a US State Department official in Kabul. “In a sense, it’s extremist nation-building.”

That’s what Nasir al-Wuhayshi, the head of Al Qaeda in the Arabian Peninsula, is hoping to do. Mr. Wuhayshi is a thin, olive-skinned Yemeni with dark hair and crooked teeth. He explains in a letter to fellow extremists that the “places under your control are a model for an Islamic state.” And he encourages them to provide basic services to locals, much like a government might do.

This type of state sounds eerily similar to what IS leaders are trying to create. IS has emerged as Al Qaeda’s premier Pan-Islamic competitor. Formerly Al Qaeda in Iraq, IS broke away from Al Qaeda in early 2014 because of a series of personality, ideological, and tactical disputes.

Its leader, Abu Bakr al-Baghdadi, could not bear coming under the control of Al Qaeda any longer. And IS’s anti-Shiite attacks and brutal executions, including beheadings and burnings, were too extreme even for Al Qaeda. But IS and Al Qaeda have a similar goal: to establish a radical Islamic emirate.

“Rush O Muslims to your state. Yes, it is your state,” says Mr. Baghdadi in a recent announcement, asking for volunteers to immigrate to Iraq and Syria to fill key positions.

IS leaders such as Baghdadi have focused most of their operations on Iraq and Syria. But they have also attempted to expand their network into Africa, other countries in the Middle East, and South Asia.

In Nigeria, for instance, the terrorist organization Boko Haram recently pledged its allegiance to IS. While the move might end up aiding the group with fundraising and recruitment, it was largely seen as a public relations stunt to help counter recent military setbacks Boko Haram has suffered at the hands of Nigerian and neighboring government forces.

In Libya, IS sent emissaries in late 2014 to meet with extremist groups across the country in an effort to establish a formal relationship. IS fighters now control key sections of Libyan cities like Surt, along the Mediterranean coast. In Egypt, leaders from the group Ansar Bayt al-Maqdis, based in the Sinai Peninsula, pledged their loyalty to IS after a series of meetings and electronic communications.

In addition, other jihadist groups, such as the various Ansar al-Sharia organizations in Libya, exist that aren’t members of either IS or Al Qaeda. The rise of these groups has forced the umbrella networks to compete more for fighters, money, and influence.

While IS and Al Qaeda both want to establish Islamic emirates, they differ in important ways. IS has a separate command-and-control structure with committees that cover the media, administrative activities, military operations, Islamic law, and other matters.

IS is also less reliant on funding from Persian Gulf donors and raises money from such activities as smuggling oil, selling stolen goods, kidnapping and extortion, and seizing bank accounts.

While both movements view Shiite Muslims as infidels, IS has conducted more attacks against Shiites than any other jihadist group. As its beheadings and burnings highlight, IS operatives have also been more inclined to conduct grisly attacks. A decade ago, Al Qaeda leader Zawahiri wrote a letter to extremists in Iraq – the predecessors of IS – warning that their gruesome practices were counterproductive.

“Among the things which the feelings of the Muslim populace who love and support you will never find palatable – also – are the scenes of slaughtering the hostages,” Zawahiri scolded.

The warnings went unheeded. And the differences between IS and Al Qaeda have turned key parts of the Islamic world into a fierce competition between the two movements. Among the most intense battlegrounds is the Horn of Africa.

• • •

Where they flourish

The heat in Djibouti is oppressive. Sun-baked, mud-brick buildings dot the country’s landscape, caked in a layer of dirt and dust. Its capital, Djibouti city, is built on coral reefs that jut into the southern entrance of the gulf. The country is strategically located on the Bab el-Mandeb Strait, which separates the Red Sea from the Gulf of Aden.

For US counterterrorism officials, Djibouti sits on a critical seam. It borders Somalia, home to the Al Qaeda-affiliated group Al Shabab. And it lies less than 20 miles from Yemen, home to Al Qaeda in the Arabian Peninsula.

“The trend is unmistakable,” says a US military official in Djibouti. “There are more violent extremists in this region than we’ve ever seen before. No comparisons.”

Take Yemen. In January, the government collapsed as Houthi rebels, a Zaidi Shiite movement from northern Yemen, took control of key ministries, and President Abdu Rabbu Mansour Hadi, an American ally, resigned. Over the past several weeks, Al Qaeda fighters have expanded their attacks across multiple provinces.

Meanwhile, Al Shabab distributed a video on Twitter recently threatening attacks against malls in the West. “What if such an attack were to occur in the Mall of America in Minnesota?” asks a masked fighter, cloaked in a checkered head scarf and wearing military fatigues. “Or the West Edmonton Mall in Canada? Or in London’s Oxford Street?”

Based on these developments, Djibouti has become a major base of operations. In 2001, the Djiboutian government reached an agreement with the United States to use Camp Lemonnier as a hub of counterterrorism activity. Since then, the US presence has grown. Camp Lemonnier now serves as the US headquarters to train, advise, and assist governments in the region in fighting extremist groups, under the command of Combined Joint Task Force – Horn of Africa. It’s also a critical node for strikes against groups in Yemen, Somalia, and other countries.

But the surge in terrorist activities isn’t just confined to the Horn of Africa. In addition to Yemen, Libya has become a breeding ground for new groups because of the collapse of its government only four years after the overthrow of Muammar Qaddafi. While Mr. Qaddafi’s demise and the July 2012 democratic elections represented a remarkable achievement for political freedom, Libya faces massive challenges.

The bureaucracy is weak, well-armed militias control much of the countryside, and extremist groups have attacked Sufi shrines across the country by digging up graves and destroying mosques and libraries. Ansar al-Sharia Libya, a loose collection of extremist groups, has emerged in this vacuum. Based in such cities as Benghazi, Darnah, and Misurata, which hug the Mediterranean coast, Ansar al-Sharia Libya seeks to establish sharia in the country.

Overall, the total number of extremist groups across the region jumped 58 percent between 2010 and 2013, according to a study by the RAND Corp. The number of extremist fighters increased dramatically, too – more than doubling between 2010 and 2013, to a high of more than 100,000 fighters.

The war in Syria is the most important attraction for fighters. Extremist groups represent a significant portion of the Syrian rebel manpower against the regime of Bashar al-Assad, including IS, Jabhat al-Nusra, Suqour al-Sham, Ahrar al-Sham, Liwa al-Islam, and Liwa al-Tawhid.

The levels of extremist violence have also grown. Among Al Qaeda affiliates alone, the number of attacks more than doubled between 2010 and 2013. But most are not directed at the US – or the West more broadly. Roughly 98 percent of these attacks targeted local regimes and civilian populations across such countries as Iraq, Syria, Yemen, and Somalia.

This rise in extremism has been caused, in part, by a growing weakness of governments across Africa and the Middle East, where the Arab uprisings created an opportunity for radicals to secure a foothold.

Since 2010, governance indicators in these areas have dropped markedly in such categories as political stability, rule of law, and control of corruption, according to World Bank data.

The surge has also been caused by the transnational movement of fighters trained on battlefields in Iraq, Syria, Afghanistan, and Pakistan. These wars have provided a unique environment for extremists to pray, share meals, train, socialize, and fight together. A growing number of these operatives have moved from these battlefields to new locations in Africa, the Middle East, and South Asia.

Not surprisingly, these trends have caused alarm in Western capitals, including London.

• • •

Risk of the returning recruit

The headquarters of Britain’s domestic intelligence agency, MI5, is nestled along the Thames River in central London. The building, called Thames House, was designed by Sir Frank Baines in the imperial neoclassical tradition. Statues of St. George and Britannia dot the building’s Portland stone facade. The Westminster coat of arms, mounted on the building, aptly reads “Custodi civitatem domine” in Latin – or “Lord protect the citizens.”

MI5 has a long history of trying to protect its population from terrorism and working closely with its American partners. Of particular concern to the agency today are Islamic extremists trained in Syria and Iraq, who also pose a threat to the US. Approximately 600 British extremists have traveled to Syria and Iraq, MI5 estimates. Many have joined IS. British agencies have watched with unease the growing number of attacks and plots across the West tied either formally or informally to Syria and Iraq.

These include attacks in Brussels in May 2014; Ottawa in October 2014; Sydney, Australia, in December 2014; Paris in January 2015; and Copenhagen, Denmark, in February 2015. More broadly, more than 20 terrorist plots in the West were either directed or provoked by extremist groups in Syria between October 2013 and January 2015, according to MI5.

“Our surveillance resources are overwhelmed,” says one British government official.

Despite the challenges, MI5 and local counterterrorism units remain aggressive. In England and Wales, terrorist-related arrests have jumped 35 percent since 2011. And more than 140 individuals have been convicted of terrorism-related offenses since 2010, according to MI5 statistics.

The British are not alone. Counterterrorism agencies across Europe and North America are under tremendous pressure to prevent attacks. A growing contingent of foreign fighters – more than 20,000 – is traveling to Syria to fight in the war, according to data collected by the US National Counterterrorism Center. Approximately 3,400 fighters, or 17 percent, appear to be coming from the West, especially from Europe.

It is difficult to predict whether most of these fighters will remain in Syria, move to future war zones in other regions, or return to the West. And even if some return, it is uncertain whether they will help hatch terrorist plots, focus on recruiting and fundraising, or become disillusioned with terrorism.

Still, foreign fighters have historically been agents of instability. Volunteering for war is often the principal steppingstone for individual involvement in more extreme forms of militancy.

And this struggle is as much about ideas as it is about military combat. It is a clash increasingly occurring online and on social media forums. Indeed, IS’s sophisticated use of social media has created opportunities for the group to reach potential recruits or influence those inspired by its message.

One of the most important forums is IS’s online magazine, Dabiq.

• • •

How dangerous, really?

The seventh issue of Dabiq, published in February, boasts a sleek cover photograph. It shows two imams, clad in creamy white robes and wearing snuggly fitting prayer caps, holding signs emblazoned with the words “JE SUIS CHARLIE” (“I AM CHARLIE”).

It is the slogan adopted by those who denounced the January attack against the French satirical magazine Charlie Hebdo.

Like all issues of Dabiq, which is printed in several languages including English, the seventh installment includes an assortment of articles intended to establish the religious legitimacy of the group and encourage extremists to come to Syria and Iraq – or else conduct attacks in their home countries.

The feature article, which accompanies the cover image, is titled “The Extinction of the Grayzone.” It starkly divides the world into two camps: Islam, represented by IS and its supporters, and the West and its followers. The article denounces Muslims that show sympathy for the victims of the Charlie Hebdo attack as apostates – guilty of abandoning Islam.

Since its expansion in Iraq and Syria, IS has become a growing threat to the US. Rather than the complex attacks on 9/11, which involved years of training and meticulous planning, the most likely IS threat today comes from smaller, less-sophisticated attacks from individuals who have taken up the cause.

“The uptick in moderate-to-small scale attacks in the West since last summer by individual extremists reinforces our assessment that the most likely and immediate threat to the Homeland will come from Homegrown Violent Extremists, or individuals with loose affiliation to terrorist groups overseas,” said Nicholas Rasmussen, director of the National Counterterrorism Center, at a US Senate hearing in February.

IS is not the only extremist group that could mount an attack on US soil. Al Qaeda in the Arabian Peninsula provided training to two operatives involved in the Charlie Hebdo shootings, Said and Chérif Kouachi. Several Yemen-based operatives continue to plot attacks against the US as well.

Core members of Al Qaeda, based in Pakistan, also present a threat to the US homeland. But their leaders have had difficulty recruiting – or even inspiring – competent operatives in the West. That’s why Zawahiri sent a small group of operatives, referred to as the Khorasan Group, to Syria to plot attacks in Europe and America.

In addition, a small number of individuals who have embraced Al Qaeda’s ideals, like the Tsarnaev brothers, who perpetrated the April 2013 Boston Marathon bombings, concern security officials. Still, terrorists have had difficulty striking the US because of robust counterterrorism steps by the Department of Homeland Security, Federal Bureau of Investigation, and other federal and local agencies. Authorities have thwarted all but four of more than 40 home-grown terrorist plots since 9/11.

Several groups pose what experts consider a medium-level threat because of their capability to target US citizens overseas, not the US homeland. Ansar al-Sharia Tunisia, for instance, has planned attacks against American diplomats and infrastructure in Tunis, including the US embassy. In Somalia, Al Shabab’s objectives are largely parochial: to establish an extreme Islamic emirate in Somalia and the broader region. But it does possess an ability to strike targets in East Africa.

Other extremist groups represent, at best, a low-level threat to the US. These groups do not possess the capability or intent to target America domestically or overseas. They include organizations such as the East Turkestan Islamic Movement, which is mainly interested in Chinese targets. Even in Afghanistan, many local groups have little interest in attacking the US homeland.

• • •

A threat overrated

While IS and other extremist groups have made some gains, in the long term they face a challenge because their firebrand version of Islam is unpopular. After the IS execution of Jordanian Air Force pilot Muath Kassasbeh by immolation in early February, a groundswell of opposition surfaced across the Muslim world.

Numerous activists on Twitter accounts and English-language jihadist forums condemned the actions as un-Islamic. They argued that burning Muslims is strictly forbidden in Islam.

“I have become very troubled upon hearing this news, because I thought that burning anyone (even animals) was not allowed under any condition in Islam,” posted one participant on the Ansar al-Mujahideen English Forum, using the name “pathoftrials.”

Islamic scholars have also been widely critical of IS. “What happened to the Jordanian pilot is by all means a crime. This barbaric action is far away from humanity, much less religion. Islam is innocent of this act,” said Sheikh Shawki Ibrahim Abdel-Karim Allam, the grand mufti of Egypt.

What’s more, support for extremist groups across the Muslim world is low, according to survey data from the Pew Research Center. Al Qaeda received negative marks in all 14 countries surveyed. In addition, the vast majority of respondents, both Muslims and Christians, have an unfavorable view of Boko Haram in Nigeria.

Concern about Islamic extremism is growing among countries with substantial Muslim populations as well. It jumped from 81 percent in 2013 to 92 percent in 2014 in Lebanon, 71 percent to 80 percent in Tunisia, and 69 percent to 75 percent in Egypt, according to the Pew Research Center.

Viewed in this context, the rise of extremist groups may well be fleeting. With little local support, they lack the foundation necessary for a sustainable movement. Even IS has had trouble holding ground on its home turf, as Iraqi government forces and local militias have retaken control of key portions of cities like Tikrit, Iraq.

Deep divisions also exist among these groups about ideology, tactics, and objectives. For all the strengths of today’s Islamic extremists, most are not committed to or capable of conducting sophisticated attacks in the West, like on 9/11.

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“Most of the plots uncovered in the United States were amateurish schemes that were detected long before they got close to being operational,” says Brian Jenkins, a terrorism expert at the RAND Corp. “Two-thirds of the US plots involved single individuals. Most of the remaining plots were tiny conspiracies. This is unlikely to change in the foreseeable future.”

Seth G. Jones is director of the International Security and Defense Policy Center at the RAND Corp., and adjunct professor at Johns Hopkins University’s School for Advanced International Studies. His most recent book is “Hunting in the Shadows: The Pursuit of Al Qa’ida Since 9/11.”