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The blast happened at about 16:20 local time (13:20 GMT) on Sunday on a shopping street in the Taksim Square area.
A suspect has now been arrested, the interior minister said.
Vice-President Fuat Oktay earlier said the blast was thought to be a terrorist attack carried out by a woman.
President Recep Tayyip Erdogan said the perpetrators would be punished.
Speaking at a news conference in Istanbul, he condemned what he called the "vile attack" and said "the smell of terror" was in the air.
Justice Minister Bekir Bozdag told Turkish media a woman had sat on a bench in the area for more than 40 minutes, leaving just minutes before the blast took place.
On Monday morning, Interior Minister Suleyman Soylu said a person suspected of having left the bomb had been arrested by police, and accused the Kurdistan Workers' Party (PKK) of responsibility.
The PKK is a militant group which calls for an independent Kurdish state within Turkey. The EU and the US regard it as a terrorist organisation.
No one has so far claimed responsibility for the blast.
Government minister Derya Yanik wrote in a tweet that a government ministry employee and his young daughter were among the victims.
BBC correspondent Orla Guerin, who is in the area, said there was a heavy police presence around Istiklal Street, which had been cordoned off. Helicopters were circling overhead as ambulances went back and forth.
Many shopkeepers standing in their doorways on the normally bustling street looked stunned, she said, adding that the incident will have come as a shock to many in the city.
Hayat, who was in an internet café on Istiklal Street when the blast took place, said there was turmoil following the explosion.
"I saw people running around and wounded people were passing by the internet café towards hospital," she said. "It was a frenzy."
Another eyewitness, Cemal Denizci, was about 50m (54 yards) from where the blast took place when it happened. "There was black smoke. The noise was so strong, almost deafening," he told AFP.
20-year-old Eyup told the BBC "there is fear" among residents of Istanbul following the attack, adding that more people may choose to stay away from crowded areas like Taksim.