Two bombings hit Baghdad on Tuesday, killing at least 44 people and wounding more than 90, police and medical sources said, following the bloodiest week of attacks inside the capital so far this year.

A suicide bombing in a marketplace in the northern district of al-Shaab killed 38 people and wounded more than 70, while a car bomb in the southern neighbourhood of al-Rasheed left six dead and another 21 wounded, the sources said.

A spokesman for Baghdad Operations Command told state television the attacker in al-Shaab, a predominately Shia Muslim area, had set off an explosives-filled vest in coordination with a planted bomb. Initial investigations revealed the attacker had been a woman, he said.

There was no immediate claim of responsibility for the attacks. Islamic State has claimed bombings in and around the capital last week that killed 100 people and sparked popular anger against the government for failing to ensure security.

Prime Minister Haider al-Abadi has said a political crisis sparked by his attempt to reshuffle the cabinet in an anti-corruption bid was hampering the fight against Islamic State and creating space for more insurgent attacks on the civilian population.

Security has improved somewhat in the capital in recent years, even as Islamic State fighters seized swathes of the country almost to the outskirts of Baghdad's ramparts.

But the prospect that the capital could return to the days when suicide bombings killed scores of people every week adds to pressure on Abadi to resolve the political crisis.