Friday's school blaze is the second fire tragedy to strike the southern Indian state of Tamil Nadu this year.
Doctors said the hospital did not have enough resources
Even after darkness had fallen, many hours after the fire had been put out, wailing parents were searching the debris looking for the bodies of children.
At least 80 children were burnt to death in the blaze, and more than 50 are in hospital, some in a critical condition.
There were scenes of chaos outside the school after the blaze broke out at around 1100 am (0530 GMT) on Friday.
The authorities say they believe the fire was caused by a spark from a thatched-roof kitchen as staff prepared the free midday meal.
At the time, some 200 children, aged between eight and 10, were in a terraced area of the building - also under a thatched roof.
Eyewitnesses said that, before the children could rush out, the roof fell on them like a ball of fire, blocking their escape down the one narrow staircase.
Firefighters used heavy equipment to smash their way through the walls of the building.
They found charred bodies lying two or three deep, some locked in an embrace, reported AFP news agency.
"Parents are howling outside the building, unable to recognise their little ones, as the bodies are being loaded into ambulances," one witness told AFP.
Television pictures showed the area where the children had attempted to flee - littered with lunch boxes, charred school books, bags and shoes.
Some of the injured children had suffered up to 80% burns.
Medical services admitted they were struggling to deal with the injured youngsters, as families and friends waited at the hospital.
"There is absolute chaos in the hospital. Parents are rushing in, trying to locate their children, " Dr P Kumar told BBC News Online from the casualty ward of Kumbakonam's state-run hospital.
"Many of them are breaking down and wailing especially as many of the bodies are burned beyond recognition."
Dr Kumar made it plain that the hospital did not have enough resources to cope with the scale of the disaster.
"We only have 21 doctors on staff. Now all the town's doctors have rushed in to help us. Help is also coming from a medical college located 40km away," he said.
A senior local official, J Radhakrishnan, echoed the sense of loss.
"Parents are looking at their dead children. It is heartrending... thousands have gathered here. It is a tragedy," he told Reuters news agency.
Angry crowds remained outside the Lord Krishna school on Friday evening, shouting slogans at the school's managers and the government, accusing them of failing to ensure adequate security measures.
Once again basic questions of fire safety in public buildings in Tamil Nadu and other parts of India will be raised.
After more than 45 people died in a fire in a marriage hall in January, the Tamil Nadu authorities promised action to improve safety measures.
But correspondents say that pledge has not been properly fulfilled.