Turkish authorities say that the bomb, which exploded late on Tuesday near a park in the city of Diyarbakir, went off as it was being transported.
The Diyarbakir governor's office said: "The bomb was made by hand, placed in a thermos and went off as it was being transported."
The information adds to suspicions that the bombing may have been a mistake, and that the device was intended for a different target.
The incident was seen as a relatively unusual attack because it was against civilians in a Kurdish area.
There was early speculation in the Turkish press that the bomb may have been intended for a police station about 1.5km away.
Police in Diyarbakir have begun a major search operation, throwing up checkpoints on main roads leading out of the city.
In a speech on Wednesday, Recep Tayyip Erdogan, Turkey's prime minister, said the details of the attack were gradually coming to light.
Erdogan said: "Our sadness is great for the victims, many of them just children, of the terror attack in Diyarbakir. The information is not yet complete, but it is slowly getting clearer."
Spate of bombings
The bombing was the deadliest in Turkey since 2003, when an al-Qaeda suicide attack in Istanbul killed 58 people.
Turkey experienced a spate of bombings in recent weeks, which have killed a total of 12 people and wounded dozens, including 10 British tourists.
The Kurdistan Liberation Hawks (TAK), the rebel group which said it carried out the previous attacks in late August, has threatened to turn Turkey into "hell".
Kurdish fighters have killed 91 soldiers and killed or injured 472 civilians so far this year, according to Turkey's foreign ministry.
At least 30,000 people have been killed in the long war between Turkey and the Kurdistan Workers' Party (PKK).