The recent gains by the insurgents have been a blow to the Shiite-led government -- as sectarian violence has escalated since the U.S. withdrawal. U.S. Secretary of State John Kerry said Washington was "very, very concerned" by the fighting but would not send in American troops.
Video of the airstrikes in Anbar -- apparently taken by aircraft at night -- was released by Iraq's Defense Ministry showing al-Qaida hideouts being bombarded. It showed men gathered around a vehicle, then running away as the site was struck.
A ministry statement said the air force struck a militants' hideout overnight, identifying them as belonging to the al-Qaida-linked Islamic State in Iraq and the Levant, which the government refers to as "terrorists."
The army and allied tribesmen also fought al-Qaida militants around the provincial capital of Ramadi on Sunday, two Anbar government officials told The Associated Press by telephone. They said 22 soldiers and 12 civilians were killed, along with an unknown number of militants, and 58 people were wounded. The officials spoke on condition of anonymity because they were not authorized to brief reporters.
Clans inside the city of Fallujah have started to form brigades, they said, and some of the factions who fought the Americans following the U.S.-led invasion a decade ago say they do not want the Iraqi army to enter the city. There was no fighting inside the city on Sunday.
Government troops, backed by Sunni tribesmen who oppose al-Qaida, have encircled Fallujah for several days, and have entered parts of Ramadi. On Friday, troops bombarded militant positions outside Fallujah with artillery, a military official said, speaking on condition of anonymity as he was not authorized to release information.
The deadliest attack Sunday in Baghdad took place in the northern Shiite Shaab neighborhood, where two car bombs exploded simultaneously near a restaurant and a tea house. Officials say those blasts killed 10 people and wounded 26.
Authorities said a car bomb ripped through the capital's eastern Shiite district of Sadr City, killing five and wounding 10. Another bombing killed three civilians and wounded six in a commercial area in the central Bab al-Muadham neighborhood, officials said. Two other bombings killed two civilians and wounded 13, police said.
Medical officials confirmed the causality figures. All officials spoke on condition of anonymity because they were not authorized to release information.
Clashes have been taking place since Monday in Ramadi and nearby Fallujah, and the Baghdad bombings could be seen as an attempt by militants to distract security forces.
Earlier on Sunday, a senior Iraqi military commander said that it will take a few days to fully dislodge al-Qaida-linked fighters in the two cities.
Lt. Gen. Rasheed Fleih, who leads the Anbar Military Command, told state TV Sunday that "two to three days" are needed to push the militants out of Fallujah and parts of Ramadi. Fleih added that pro-government Sunni tribes are leading the operations while the army only is offering aerial cover and logistics on the ground. He didn't elaborate on the operations.
"The quiet and safe life that is sought by the Anbaris will not be completely restored before few hours or two to three days, God willing," Fleih said.
Ramadi was a stronghold of Sunni insurgents during the U.S. war. Al-Qaida militants largely took both cities over last week and have been fending off incursions by government forces there since.
ISIL is also one of the strongest rebel units in neighboring Syria, where it has imposed a strict version of Islamic law in territories it holds in the civil war raging there. It also has kidnapped and killed dozens of people it deems critical of its rule. On Saturday, it claimed responsibility for a suicide car bombing in a Shiite-dominated neighborhood in Lebanon.
Tensions in Anbar have run high since Dec. 28, when Iraqi security forces arrested a Sunni lawmaker sought for terrorism charges. Two days later, the government dismantled a months-old, anti-government Sunni protest camp, sparking clashes with militants.
To ease the tension, the government of Prime Minister Nouri al-Maliki withdrew army forces from the cities. Sunni lawmakers see the army as a tool of al-Maliki to target his rivals and consolidate power.
Kerry said the U.S. would support Iraq in its battle with al-Qaida, but without sending in troops.
The U.S. was "very, very concerned" by the fighting, he told reporters as he left Jerusalem for Jordan and Saudi Arabia on Sunday to discuss his effort to broker peace between Israel and the Palestinians. However, he said the U.S. would not send ground troops in to support Iraqi forces.
The U.S. recently supplied Iraq with Hellfire missiles often used against militants.
Violence in Iraq spiked in April after the government staged a crackdown on a major Sunni protest camp. Iraq's al-Qaida branch has fed on Sunni discontent and on the civil war in neighboring Syria, in which mostly Sunni rebels fight a government whose base is a Shiite offshoot sect.
Militants have targeted civilians, particularly in Shiite areas of Baghdad, with waves of coordinated car bombings and other deadly attacks.
According to the U.N., Iraq had the highest annual death toll in 2013 since the worst of the country's sectarian bloodletting began to subside in 2007. The U.N. said violence killed 8,868 last year.
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