The blasts occurred nearly simultaneously 200m apart near a telephone exchange in the Talbiya district of Baghdad on Monday, police told Reuters.
The violence began when a car parked near a repair shop exploded, The Associated Press reported.
There were conflicting reports about whether the second blast was caused by a suicide bomb or mortar fire.
The area is a bastion of al-Mahdi Army militia, led by cleric Moqtada al-Sadr.
Shortly afterwards, a bomb planted outside a restaurant near the central bank in central Baghdad killed six and wounded 28, and the toll was expected to rise, police said.
The bombs came a day after Shia gunmen rampaged through a Sunni district, killing at least 40.
Those killings, the worst of their kind in Baghdad, came after a car bomb attack on a Shia mosque in the Jihad area on Saturday evening and were followed by a double car bombing at another Shia mosque late on Sunday that killed 19.
Other bombings and attacks took place across Baghdad and the rest of Iraq.
A roadside bomb struck a police patrol near a restaurant in eastern Baghdad, wounding three policemen, police Lieutenant Ahmad Qassim said.
In Kirkuk, a suicide truck bomb struck an office of the Patriotic Union of Kurdistan, one of the main Kurdish political parties in Iraq, killing five people and wounding 12 others, police brigadier Sarhat Qadir said.
A police patrol in the predominantly Shia city of Hillah, about 95km (60 miles) south of Baghdad, hit a roadside bomb, leaving one policeman dead and four wounded, army captain Hassim al-Khafaji said.