The mayor criticised the response of security forces after the blast.
"Why did it take security forces more than half-an-hour to get to the area?" he said.
"Why did they not send us more support than just some ambulances and firemen?"
Crowds filled the main square and held up body parts while chanting anti-American slogans and condemning attacks.
One man at the scene, Hussein Kadhim, said: "We blame the Iraqi government and the American army for this incident."
Ahmed Mohammed asked: "How could a driver leave his car in this area? How could he have passed through several checkpoints and then park it in a forbidden zone."
Kufa is a stronghold of supporters of Muqtada al-Sadr, a Shia leader, and each time government security forces fail to prevent an attack it triggers calls for his Mahdi Army militia to return to the streets.
Kufa, 160km south of Baghdad, is near the Shia city of Najaf, one of the holiest places in Shia Islam.
Last month, a suicide car bomber killed 60 people and wounded 170 near a shrine in Karbala, also in the Shia south of Iraq.
A US-backed security crackdown in Baghdad has reduced the number of sectarian killings blamed on death squads, but a string of car bombs has killed hundreds in recent weeks.