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.

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 so-called Islamic State of Iraq and the Levant (ISIL) terrorist group.

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, while Katyusha rockets also hit the heavily fortified Green Zone in central Baghdad, home to parliament and the US embassy, officials said.

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 militant groups including ISIL are typically blamed for violence in the capital.

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