Another 91 people have been wounded in the attack, which took place in the central Rusafa district at sunset.
The first blast appeared to target an Iraqi army vehicle on patrol, a security official said.
The second blast erupted immediately after the first with the third a few moments later, rocking windows 500 metres away on the opposite bank of the river Tigris.
Shortly afterwards gunfire erupted around the city.
Güncelleme Tarihi: 20 Eylül 2018, 18:16
Ten shops were burnt out in the blasts near Al-Wathba square, a historic commercial area popular with Shia Kurds. An interior ministry official warned that the death toll was expected to rise.
Around an hour after the bomb detonations, five more explosions were heard. The blasts were thought to be mortars fired by one of the city's warring armed groups.
Saturday's attack followed a major security operation on Friday in the nearby Fadhel neighbourhood, when Iraqi and US forces raided homes and arrested 28 people.
In a separate incident, five Iraqi soldiers returning to their base north of Baghdad in civilian clothes were killed when fighters sprayed their cars with automatic fire, Ali Mohammed Hassan of the Balad police said.
Meanwhile, Iraqi and US military officials said that between 30 and 35 suspects had been arrested in raids in the city of Baquba, northeast of Baghdad.
An Iraqi army major, speaking on condition of anonymity, said that two of the detained suspects were Egyptians who had joined the Sunni Arab insurgency against the Shia-led government and its US backer.