Story highlights Prime Minister Haider al-Abadi: Sunnis "almost on board" with Shiite-led government

He says all Iraqis "must work together," be willing to make tough decisions

Al-Abadi says his government has been warning about ISIS "bloodbath" for years

He hopes airstrikes don't lead to "the rise of another terrorist element instead of" ISIS

Iraq's prime minister said Tuesday he is happy to see the United States and its Arab allies striking ISIS targets in Syria, so long as they "do it right this time."

Prime Minister Haider al-Abadi told CNN's Christiane Amanpour that Iraq has "paid a heavy price" for polarization in the region and instability in Syria, saying ISIS fighters have inflicted tremendous pain, suffering and losses on his people after crossing into Iraq. He expressed hope the airstrikes will do what U.S. President Barack Obama has vowed: degrade and destroy the group, which calls itself the Islamic State.

It's good some Arab nations have joined the American-led military campaign, al-Abadi added, though he said he wished they had understood and acted on the danger posed by ISIS sooner.

"We have warned ... this is going to end in a bloodbath if nobody stops it," he said. "Nobody was listening."

The newly installed prime minister said, "I personally am happy that everybody is seeing this danger, so that they are going to do something about it. And I hope they ... do it right, and they don't do it their own way."

Photos: U.S., Arab nations attack ISIS in Syria Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – In this photo released by the U.S. Air Force, fighter jets fly over northern Iraq as part of coalition airstrikes in Syria on Tuesday, September 23. The United States and several Arab nations have started bombing ISIS targets in Syria to take out the militant group's ability to command, train and resupply its fighters. Hide Caption 1 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – A U.S. Navy jet is refueled in flight after airstrike missions in Syria on September 23. In addition to bombing ISIS, the United States has also taken action -- on its own -- against another terrorist organization, the Khorasan Group. U.S. President Barack Obama described Khorasan members as "seasoned al Qaeda operatives in Syria." Hide Caption 2 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – This photo, released by activists in Idlib, Syria, on September 23, shows what they say is the aftermath of U.S. airstrikes against ISIS. Hide Caption 3 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – Another aftermath image from the Idlib activists. Hide Caption 4 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – Obama, speaking at the White House on September 23, addresses the ISIS airstrikes. "The strength of this coalition makes it clear to the world that this is not America's fight alone," Obama said. Hide Caption 5 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – The USS Philippine Sea, a guided-missile cruiser with the U.S. Navy, launches a Tomahawk cruise missile from the Persian Gulf during an attack on ISIS positions on September 23. The U.S. Navy released this photo and several others following the airstrikes. Hide Caption 6 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – U.S. sailors stand on the bridge of the USS Philippine Sea while Tomahawk cruise missiles are launched toward targets in Syria on September 23. Hide Caption 7 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – The USS Arleigh Burke, a guided-missile destroyer with the U.S. Navy, launches Tomahawk missiles in the Persian Gulf on September 23. Hide Caption 8 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – An F/A-18C Hornet prepares to take off from the flight deck of the USS George H.W. Bush, a U.S. Navy aircraft carrier in the Persian Gulf, before conducting strike missions against ISIS targets on September 23. Hide Caption 9 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – Tomahawk missiles fired by the USS Philippine Sea fly in the air above the Persian Gulf on September 23. Hide Caption 10 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – Two more U.S. fighter jets prepare to take off from the USS George H.W. Bush and conduct airstrikes against ISIS targets in Syria. Hide Caption 11 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – This before-and-after image, provided by the U.S. Department of Defense, shows what is purported to be an ISIS command center that was targeted by airstrikes in Raqqa, Syria. Hide Caption 12 of 13 Photos: U.S., Arab nations attack ISIS in Syria U.S., Arab nations attack ISIS in Syria – The Defense Department also released this before-and-after image of a purported ISIS finance center in Raqqa. Hide Caption 13 of 13

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Elaborating on that latter point, al-Abadi said, "I hope we're not going to see a crush of DAIISH" -- another term for ISIS that some say sounds like an Arabic word for "to trample on," which the militants reportedly despise -- "and the rise of another terrorist element instead of them."

One thing the U.S. military has not done well enough when it comes to its strikes against ISIS targets in Iraq is work in close coordination with Iraqi ground forces, according to al-Abadi.

"Our forces are moving forward and, when they are moving forward, they need air cover, they need air support," said the prime minister. "That air support is not forthcoming."

Al-Abadi added: "We ask for it, but of course the coalition -- they have their own ... plan."

'If we don't work together, we don't deserve a country'

That comment may beg the question: What is al-Abadi's plan for Iraq?

The Middle Eastern nation has been in turmoil for years. In addition to questions about its effectiveness in providing security and basic needs to its people, the government was panned for being dominated by Shiite Muslims -- like former Prime Minister Nuri al-Maliki -- and alienating Sunnis. (Notably, ISIS is made up of Sunnis pushing to institute their hard-line brand of Sharia law in any territory they control, while ruthlessly lashing out at those who don't subscribe to their mission.)

After Obama's withdrawal of U.S. ground troops, Iraq's military was maligned, especially after ISIS fighters speedily overran wide swaths of the country, at times prompting Iraqi troops to drop their weapons and run. Al-Abadi conceded Tuesday that the Islamist extremist group controls at least 25% of land in his country and is on the doorstep of the capital, Baghdad.

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This isn't to say al-Abadi deserves blame for this predicament. He officially took power on September 8, a few weeks after being endorsed by al-Maliki, his fellow Shiite and Islamic Dawa Party member.

Still, even if he hasn't been in charge long, this is al-Abadi's problem now.

The new prime minister said he does not want militias any more in Iraq, saying armed forces "must all be under the umbrella of the state." Al-Abadi added that he prefers that those in ISIS-controlled territories "liberate their own areas" with support from the government, rather than have Iraqi troops march in.

As to bringing the country together, the Iraqi leader said, "We've worked hard to make (the government) more inclusive" and "I think we've got (Sunnis) almost on board right now." Al-Abadi said a priority for him after he leaves New York -- where he, like other world leaders, are at the U.N. General Assembly -- will be nominating defense and interior ministers, one of whom will be Sunni.

Its history of secular and political division, not to mention bloodshed, notwithstanding, al-Abadi said Iraq can survive if Shiites and Sunnis can work together.

"I think we can. We have to," he added. "... This is our country. And if we don't work together, we don't deserve a country.

"... We have to work together. We have to give in to each other. And I'm prepared to do hard decisions. ... I expect others to do their own as well."