The first blast occurred in a rubbish bin in the busy Gungoren residential area. The second, larger explosion occurred as crowds gathered.
No group has claimed responsibility, but security services said the attack bore the hallmarks of Kurdish rebels.
President Abdullah Gul said the attack showed “the ruthlessness of terrorism”.
After the explosions, there were scenes of panic, with people covered in blood as they tried to run from the scene. TV footage showed many victims lying on the street and being carried to ambulances in blankets.
Initial reports suggested it may have been a gas leak, but Istanbul Governor Muammer Guler later said he was “certain that this is a terror attack” aimed at causing maximum casualties.
The blasts occurred about ten minutes apart around 2200 local time (1900 GMT) on a busy pedestrian street.
The editor of the New Anatolian newspaper, Ilnur Cevik, told the BBC that about 1,000 people had been in the area at the time.
The first explosion was caused by a small device placed in a rubbish bin in front of a bank. Afterwards, a crowd gathered in the area.
Then a second bomb placed in another bin about 50m from the first exploded minutes later.
“The first explosion was not very strong,” Huseyin Senturk, the owner of a nearby shoe shop, told the Associated Press.
“Several people came to see what was going on. That”s when the second explosion occurred and it injured many onlookers.”
Mr Guler said police believed the blasts were not suicide bombings, but activated either remotely or by means of a timer.
“This is an abhorrent attack. Unfortunately, the fact that the explosion took place in a crowded place increased the number of the casualties,” he said.
“An extensive investigation is being conducted at present at the scene of the incident.”
Turkish media quote police sources as saying the attack bears the hallmarks of the rebel Kurdistan Workers Party (PKK), possibly in revenge for a series of major operations by the Turkish military on its bases in recent days.
The BBC”s Sarah Rainsford in Istanbul says the PKK has bombed civilians in Turkish cities in the past, but so have Islamist militants and other radical groups.
In November 2003, more than 60 people were killed by a series of suicide bombings in Istanbul which the authorities linked to al-Qaeda.
Kurdish rebels carried out a spate of attacks on tourist sites in Turkey in 2006, killing more than a dozen people.
As the police investigation continues, Turkish politicians have condemned Sunday”s attack.
President Gul said: “I condemn the perpetrators of this attack which demonstrates the ruthlessness of terrorism and its goal to engage in savagery without any regard for women, men, the elderly or children.”
Prime Minister Recep Tayyip Erdogan such attacks only strengthened Turkey”s determination in its fight against terrorism.
A politician from Mr Erdogan”s governing AK Party, Murat Mercan, told the BBC that Turkey would not give in to terrorists.
“Terrorists are trying to destabilise the country, but Turkey has already a lot of experience on this terrorism so it won”t distract our country, our society from daily, ordinary life,” he said.
Earlier this month, three policemen and three gunmen were killed in a gun battle outside the US consulate in a northern suburb of Istanbul.
Police said they believed the attackers were members of a Turkish Sunni fundamentalist group, the Great Eastern Islamic Raiders Front.
