SOUTHERN OUTSKIRTS OF FALLUJA, Iraq (Reuters) - The Iraqi army stormed to the southern edge of Falluja under U.S. air support on Monday and captured a police station inside the city limits, launching a direct assault to retake one of the main strongholds of Islamic State militants.

A Reuters TV crew about a mile (about 1.5 km) from the city’s edge said explosions and gunfire were ripping through Naimiya, a largely rural district of Falluja on its southern outskirts.

An elite military unit, the Rapid Response Team, seized the district’s police station at midday, state TV reported.

The unit advanced another mile northward, stopping about 500 meters (yards) from the al-Shuhada district, the southeastern part of city’s main built-up area, army officers said.

The battle for Falluja is shaping up to be one of the biggest ever fought against Islamic State, in the city where U.S. forces waged the heaviest battles of their 2003-2011 occupation against the Sunni Muslim militant group’s precursors.

Falluja is Islamic State’s closest bastion to Baghdad, and believed to be the base from which the group has plotted an escalating campaign of suicide bombings against Shi’ite civilians and government targets inside the capital.

As government forces pressed their onslaught, suicide bombers driving a car and a motorcycle blew themselves up in the capital. Along with another bomb planted in a car, they killed more than 20 people and injured more than 50 in three districts of Baghdad, police and medical sources said.

Separately, Kurdish security forces announced advances against Islamic State in northern Iraq, capturing villages from militants outside Mosul, the biggest city under militant control.

The Iraqi army launched its operation to recover Falluja a week ago, first by tightening a six-month-old siege around the city 50 km (30 miles) west of Baghdad.

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Falluja, in the heartland of Sunni Muslim tribes who resent the Shi’ite-led government in Baghdad, was the first Iraqi city to fall to Islamic State in January 2014. Months later, the group overran wide areas of the north and west of Iraq, declaring a caliphate including parts of neighboring Syria.

On Monday, army units were “steadily advancing” to Falluja’s southern outskirts under air cover from a U.S.-led coalition helping to fight against the militants, according to a military statement read out on state TV.

A Shi’ite militia coalition known as Popular Mobilisation, or Hashid Shaabi, was seeking to consolidate the siege by dislodging militants from Saqlawiya, a village just to the north of Falluja.

The militias, who took the lead in assaults against Islamic State in other parts of Iraq last year, have pledged not to take part in the assault on the mainly Sunni Muslim city itself to avoid aggravating sectarian strife.

Between 500 and 700 militants are in Falluja, according to a U.S. military estimate. The U.S.-led coalition conducted three air strikes near Falluja over the past 24 hours, destroying fighting positions, vehicles, tunnel entrances and denying the militants access to terrain, it said in a statement.

ISLAMIST MILITANT STRONGHOLD

Falluja has been a bastion of the Sunni insurgency that fought both the U.S. occupation of Iraq and the Shi’ite-led Baghdad government that took over after the fall of dictator Saddam Hussein, a Sunni, in 2003.

American troops suffered some of their worst losses of the war in two battles in 2004 to wrest Falluja back from Al Qaeda in Iraq, the insurgent group now known as Islamic State.

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The latest offensive is causing alarm among international aid organizations over the humanitarian situation in the city, where more than 50,000 civilians remain trapped with limited access to water, food and health care.

Falluja is the second-largest Iraqi city still under control of the militants, after Mosul, their de facto capital in the north that had a pre-war population of about 2 million.

It would be the third major city in Iraq recaptured by the government after Saddam’s home town Tikrit and Ramadi, the capital of Iraq’s vast western Anbar province.

Falluja is also in Anbar, located between Ramadi and Baghdad, and capturing it would give the government control of the major population centers of the Euphrates River valley west of the capital for the first time in more than two years.

On the northern front, the security forces of the autonomous Kurdish region launched an attack on Sunday to oust Islamist militants from villages about 20 km (13 miles) east of Mosul so as to increase the pressure on Islamic State and pave the way for storming that city.

The Kurdish forces, known as peshmerga, have retaken six villages in total since attacking Islamic State positions on Sunday with the support of the U.S.-led coalition, the Kurdistan Region Security Council said on Monday. That represents most of the targets of their latest advance.

Prime Minister Haider al-Abadi hopes to recapture Mosul later this year to deal a decisive defeat to Islamic State.

Abadi announced the onslaught on Falluja on May 22 after a spate of bombings that killed more than 150 people in one week in Baghdad, the worst death toll so far this year. The worsening security in the capital has added to political pressure on Abadi, struggling to maintain the support of a Shi’ite coalition amid popular protests against an entrenched political class.

Monday’s bombings targeted two densely populated Shi’ite districts, Shaab and Sadr City, and a government building in one predominantly Sunni suburb, Tarmiya, north of Baghdad.

A car bomb in Shaab killed 12 people and injured more than 20, while in Tarmiya eight were killed and 21 injured by a suicide bomber who pulled up in a car outside a government building guarded by police. In Sadr City, a suicide bomber on a motorcycle killed three people and injured nine.

The battle of Falluja is helping Abadi refocus the attention of Iraq’s unruly political parties on the war against Islamic State, so as to defuse popular unrest prompted by delays in a planned reshuffle of the cabinet to help root out corruption.

In a speech to parliament on Sunday, he called on political groups to “put on hold their differences until the military operations are over.”

Washington says Islamic State’s territory is steadily being rolled back both in Iraq and in Syria, where it has lost ground to U.S.-backed, mainly Kurdish insurgents in the north and to the Russian-backed forces of President Bashar al-Assad.