The Sri Lankan army says more than 2,000 civilians have waded across a lagoon to flee the war zone in the north-east of the country.
A senior military officer said another 2,000 people were waiting to cross into government-held areas.
The reports come hours after Colombo rejected international calls to stop its offensive against the Tamil Tigers.
The UN Security Council has expressed “grave concern” at the “worsening humanitarian crisis” in the region.
It says about 50,000 people are trapped in the government-designated “safe zone” where the rebels are now surrounded by the army.
A military spokesman told the BBC that unmanned aircraft had filmed more than 2,000 people wading across the lagoon which borders the fighting zone on the non-seaward side.
Brig Udaya Nanayakkara said the civilians had braved rebel fire to reach government-held areas.
“There is a large number of people crossing, and the (rebels) fired at them. Four people were killed, 14 were wounded,” he said.
The Sri Lankan army’s version of events cannot be independently verified and there has been no comment from the rebels.
The BBC’s Charles Haviland, in Sri Lanka, says there is another unverified report that medical staff have abandoned the main hospital in the rebel-held area because of persistent shelling.
Dozens of civilians have been reported killed in artillery attacks on the facility in recent days.
The authorities and the rebels blame each other for civilian deaths.
A spokeswoman for the Red Cross (ICRC) said fighting in the area was intense and uninterrupted.
As the fighting continued, Britain said on Thursday that it supports an early inquiry into whether war crimes have been committed in Sri Lanka.
“We would support an early investigation into all incidents that may have resulted in civilian casualties… to determine whether war crimes have been committed,” said junior foreign minister Bill Rammell.
Earlier, the UN Security Council asked the Sri Lankan government and the Tamil Tigers to ensure the safety of civilians.
A statement said council members “strongly condemn the LTTE [Liberation Tigers for Tamil Eelam] for its acts of terrorism over many years”.
It urged the group to “lay down its arms and allow the tens of the thousands of civilians to leave”.
US President Barack Obama also urged the army to stop shelling of civilian areas and called on the rebels to lay down their arms.
“Without urgent action this humanitarian crisis could turn into a catastrophe,” he said.
But the government in Colombo said it would not halt the offensive because it would give the rebels time to recover.