'Ball of fire'
The attacker got off his vehicle and managed to escape before the bomb went off, an official said.
Najim Ali, a 30-year-old local who was shopping in the market, said: "I heard a boom and saw a ball of fire.
"I saw cars flying in the air because of the force of the explosion."
Saif Mohammed, a 20-year-old Sadr City resident, said: "Explosions like this confirm that the Iraqi security forces are not able to protect the people from violence or war."
Violence has dropped markedly in Iraq in recent months, with May seeing the lowest Iraqi death toll since the 2003 US-led invasion, but there are fears of a dramatic increase as US troops pull out of cities and major towns by June 30.
Three school students died in another bombing in Sadr City on Monday, one of a string of blasts that killed 27 people across Iraq that day.
At least 73 people died in a suicide lorry bombing outside a mosque in Kirkuk province on Saturday.
Following Wednesday's attack, Jalal Talabani, the Iraqi president, said that "this cowardly act will not shake the determination of our people and armed forces to take over security responsibility and defeat terrorist schemes".
Nuri al-Maliki, the prime minister, warned earlier this month that attacks were likely to increase in the coming weeks as anti-government groups attempt to undermine confidence in the Iraqi security forces, and urged Iraqis not to lose heart if attacks increase.
Jeff Morrell, a spokesman for the Pentagon, said US forces had "been alerted to the possibility we will likely see an uptick in violence leading up to the June 30 deadline".
But he added: "Despite the fact that we have seen sporadic high profile attacks still taking place in Iraq the overall security climate is a good one."
'Work to be done'
Raed Jarrar, a Washington-based political analyst, told Al Jazeera that the US withdrawal must go ahead despite people's concerns, but there was still much work to be done to stabilise Iraq.
"The US withdrawal is a necessity and Iraq is more than ready to end the occupation, but of course that deosn't mean that Iraq is in a perfect situation yet," he said.
"There is a lot of work still to be done to enhance the Iraqi security forces, to enhance the Iraqi political process and draw in all Iraqis around the country."
A US military spokesman said on Wednesday that a small number of US troops would be left in some Iraqi cities after the June 30 deadline at so-called Joint Security Stations to train and advise local security forces.
The US military will also continue to provide intelligence and air support, and be on call if needed.