The Islamic State claimed responsibility for two separate suicide bombings that killed dozens of people in Baghdad on Thursday, as the group battles an air and ground offensive targeting their stronghold of Fallujah.

A car laden with explosives targeted a busy commercial street of in the east of the capital, killing 17 people and wounding over 50, according to an Iraqi police officer.

In a second attack, a man wearing an explosive belt blew himself up at checkpoint near the barracks of Taji, just north of Baghdad, killing seven soldiers and wounding more than 20, he said.

IS claimed responsibility for both attacks on social media shortly after.

Iraqi forces, aided by US airstrikes, began their offensive against Falluja on May 23 after a series of deadly bombings hit Shi'ite districts of the capital. The ground troops entered the city for the first time on Wednesday, after surrounding it last week.

Islamic State "has a long experience in establishing small multiple networks that have the ability to operate independently from each other," said Baghdad-based analyst and former army general Jasim al-Bahadli.

Fallujah is a historic bastion of the Sunni insurgency, first against the US occupation of Iraq, in 2003, and then against the Shi'ite-led authorities that took over the country.