Iraqi forces backed by tribesmen battled jihadists on Thursday after they seized parts of two Sunni-majority cities following days of violence triggered by the demolition of a year-old protest camp.

Militants from the Al-Qaeda-linked Islamic State of Iraq and the Levant (ISIL) seized half of the city of Fallujah, outside Baghdad, and parts of Anbar provincial capital Ramadi farther west.

But security forces backed by loyalists among the province's powerful tribes launched a counter-attack, sparking fierce fighting.

"We entered Fallujah with heavy clashes," special forces commander Major General Fadhel al-Barwari said in a statement.

In Ramadi, police backed by tribal auxiliaries battled militants in the east of the city for several hours before the fighting subsided with the militants still in control of some areas.

Staff Lieutenant General Mohammed al-Askari told AFP that some of the tribesmen fighting alongside the security forces had received arms from the government.

Clashes erupted in the Ramadi area on Monday as security forces tore down the main protest camp set up by Sunni Arabs a year ago.

The violence spread to Fallujah and troops withdrew from parts of both cities, which were hubs of the insurgency that followed the US-led invasion of 2003.

Earlier on Thursday, an interior ministry official said that ISIL militants controlled half of Fallujah and parts of Ramadi, with armed tribesmen holding other areas of the two cities.

In Fallujah, militants set up checkpoints in the centre and south of the city, each manned by six to seven fighters, a witness said.

In Ramadi, an AFP correspondent saw dozens of trucks carrying heavily armed men in the east of the city, playing songs praising ISIL.

Lyrics included "The Islamic State remains" and "Our State is victorious," while militants carried black flags of a type frequently flown by ISIL.

The unrest has led to hardship in Ramadi.

"We are not leaving our homes because of what is happening," said resident Abdel Nasser.

"There is no food. Even if you manage to go to the market, you find nothing."

Police stations torched

On Wednesday, there were sporadic clashes in Ramadi, with militants torching four police stations.

In Fallujah, several police stations were torched after police abandoned most of their positions.

Militants also seized police weapons and freed more than 100 prisoners, officers said.

On Thursday, the interior ministry demanded that the deserters return.

"The interior ministry calls for all officers and policemen working in Anbar province who have left their duties" to return "immediately and do their patriotic duty in confronting the terrorist attack," a statement said.

On Tuesday, Prime Minister Nuri al-Maliki had said troops would leave Anbar's cities, but he reversed that decision on Wednesday and said he would send in reinforcements.

Security officials said on Thursday that regular army troops remained outside Ramadi and Fallujah.

But a witness in Fallujah said tanks had deployed near the city's eastern entrance.

Maliki had long sought the removal of the protest camp, calling it a "headquarters for the leadership of Al-Qaeda," but doing so has resulted in a sharp decline in the security situation.

While the camp's closure has removed a physical sign of Sunni Arab grievances, their complaints of being marginalised by the Shiite-led authorities and unfairly targeted by security forces remain unaddressed.

There has also been political fallout, with 44 MPs, most of them Sunni, announcing on Monday that they had submitted their resignations. They also called for "the withdrawal of the army from the cities and the release of MP Ahmed al-Alwani."

Alwani, a Sunni Arab and leading supporter of the protest camp, was arrested in a Saturday raid on his Ramadi home in which his brother, five guards and a security forces member died.

Protests first broke out in Sunni Arab-majority areas of western, central and northern Iraq in late 2012, and have continued for more than a year.

Attacks also hit Baghdad, Baladruz northeast of the capital, and the northern city of Mosul on Thursday, killing at least 24 people.

The deadliest attack was in Baladruz, where a suicide bomber detonated an explosives-rigged vehicle near car dealerships, killing 13 people and wounding 27.