The UN says that children are bearing the brunt of the conflict
A growing number of children have been killed or injured in Sri Lankan fighting over the last 10 days, the UN children's agency (Unicef) says.
It has called on the government and Tamil Tiger rebels to give "absolute priority" to the safety of children and the wider civilian population.
The Tamil Tigers say 250,000 civilians in the conflict zone want to stay in rebel-held areas for protection.
However the government has appealed to the Tigers to allow them to leave.
It has offered safe passage to the civilians, but President Mahinda Rajapaksa has ruled out a ceasefire as his forces continue to attack dwindling rebel territory.
Rebel political chief B Nadesan told the BBC that Tamils "do not wish to end up in the hands of their killers.
"They believe their security lies in the area under the control of our organisation - and regard the area under our control as their safe haven," he said.
Health officials and human rights groups say hundreds of civilians have died.
'Caught in crossfire'
Unicef said the safety of children - some just months old - and the wider civilian population was of paramount importance.
Sri Lankan soldiers captured Mullaitivu last Sunday
"We have clear evidence that children are being caught in the crossfire, and that children are being injured and killed," Unicef Regional Director for South Asia, Daniel Toole, said.
"It is crucial that safe areas, schools and medical facilities are protected and considered zones of peace, in all circumstances.
"Children are bearing the brunt of a conflict which is not theirs. We are gravely concerned for the tens of thousands of children who are trapped in a fast shrinking area of intense conflict."
But Sri Lanka's defence secretary, Gotabaya Rajapakse, said the numbers were exaggerated and aid agencies were panicking.
Meanwhile, the UK has announced it is doubling its emergency humanitarian aid to try to protect civilians.
President Rajapaksa said that the rebels were refusing to let the civilians leave.
"I urge the [Tamil Tigers], within the next 48 hours to allow free movement of civilians to ensure their safety and security. For all those civilians, I assure a safe passage to a secure environment," he said in a statement on Thursday.
The European Union on Friday called for a halt to the conflict.
EU Humanitarian Aid Commissioner Louis Michel said: "This is an escalating humanitarian catastrophe. We are extremely worried about the terrible situation facing people trapped in the fighting."
But the Sri Lankan government has said it will continue "to liberate areas which have not been liberated so far".
Our correspondent says that displaced civilians who do manage to leave the war zone are held in government-managed camps to which there is no media access.
The UN High Commissioner for Human Rights, Navi Pillay, has also said she is extremely concerned about the well-being of people caught up in the fighting.
She said the situation could be worse than generally realised because of the restrictions on access to the war zone.
The army is facing monsoon conditions in its push into the jungle
Ms Pillay said there appeared to be "very grave breaches of human rights by both sides in the conflict and it is imperative that we find out more about what exactly has been going on".
On Friday, the Reporters without Borders group also appealed to President Rajapaksa to allow local and foreign journalists to report freely.
The Red Cross says the humanitarian situation in the north-east "remains precarious for thousands".
"Stocks have been depleted and sustainable ways of producing food locally have become almost nonexistent," it said.
The UK has said it is doubling its emergency humanitarian aid with another £2.5m to support Red Cross operations and help maintain relief convoys.
On Thursday, aid agencies said they had evacuated hundreds of wounded civilians, including 50 critically ill children, to a hospital in the town of Vavuniya.
The military says it is involved in a final push against the retreating rebels.
It has captured the key towns of Kilinochchi, Mullaitivu and the strategically important Elephant Pass in recent weeks.
The BBC's Ethirajan Anbarasan is at Elephant Pass with the army.
He says he can hear artillery fire 15km (nine miles) to the south and has been told there is heavy fighting there.