"There was a huge explosion and before I went out to look another bomb went off," said Fadel Hussein, a waiter at a teahouse near the scene.
"Heavy smoke was everywhere. There were so many bloody victims on the ground, we helped to evacuate those people to ambulances," he told the AFP news agency.
Al Jazeera sources said 68 people were wounded in the blasts.
A doctor said several women and children were among the casualties.
Witnesses said the attack took place on a street lined with restaurants as a bus carrying children to a school drove past.
"Innocent and simple people were gathering to have breakfast or shop in the nearby area. A minibus which was driving past was also hit and four or five of its passengers were killed," Jassim Mohammed, another witness, said.
"How can you explain this act? This is not a military unit, not a military barracks. There is nothing there."
Adhamiyah, a Sunni neighbourhood in a mostly Shia eastern half of the city, saw fierce clashes at the height of Iraq's sectarian violence. Attacks in the area have sharply dropped over the last year.
Despite the improvement in security in large swathes of Iraq, including the capital, armed groups continue to launch near daily attacks, most of them targeting US and Iraqi security forces.
In another attack, in Baquba, capital of volatile northern Diyala province, a teenager in a suicide bomb vest blew herself up at a checkpoint of US-backed security personnel, killing six people and wounding 18 other.
Police said the bomber was a 13-year-old girl.