Attacks in and around Baghdad killed seven people on Tuesday as Iraqi forces made steady progress against militants in conflict-hit Anbar province where the government lost key territory weeks ago.

The bloodshed comes after more than 1,000 people were killed in January, the worst monthly death toll in nearly six years, as security forces grapple with frequent attacks and battles in Anbar with anti-government fighters, including those loyal to the powerful Islamic State of Iraq and the Levant jihadist group.

Four bombings in and near the capital killed seven people while Katyusha rockets also hit the heavily-fortified Green Zone in central Baghdad, home to parliament and the US embassy, officials said.

A car bomb in western Baghdad ripped through a market and killed four people, while blasts elsewhere in the city killed two others. Another bombing just north of capital in Taji killed a policeman.

On Tuesday, two rockets also exploded in the walled-off Green Zone complex, but caused no casualties. It was unclear what the target of the attack was.

It came a day after violence in and around Baghdad killed 28 people.

No group immediately claimed responsibility for the latest attacks, but Sunni militant groups including ISIL are typically blamed for violence in the capital.

ISIL has also been involved in fighting security forces in Anbar province, a mostly Sunni desert region bordering Syria where militants have for weeks held parts of Ramadi and all of Fallujah, which lies on Baghdad's doorstep.

On Tuesday, security forces took control of several Ramadi neighbourhoods after days of heavy clashes in the city, army chief Staff General Ali Ghaidan Majeed told AFP.

The defence ministry announced on Saturday that warplanes and artillery had hit a neighbourhood of northern Fallujah, a rare military operation inside the city itself.

A security official told AFP that an assault on the city was imminent, but a journalist in Fallujah said it was mostly calm on Monday.

The army has largely stayed out of Fallujah, a short drive from Baghdad, fearing major incursions could ignite a drawn-out campaign with high civilian casualties and heavy damage to property.

US battles in the city, a bastion of militants following the 2003 US-led invasion, were among their bloodiest since the Vietnam War.

Along with ISIL, other militant groups and anti-government tribes have fought forces loyal to the central government.

The stand-off has prompted more than 140,000 people to flee their homes, the UN refugee agency said, describing it as the worst displacement in Iraq since the peak of the sectarian conflict.