(Sri Lankan Army)
The U.N. and others have called for a negotiated truce to allow civilians to leave the rebel-held coastal strip -- and the government says more than 52,000 had escaped since Monday.
But it has refused to heed those international pleas to halt the fighting, saying it is on the verge of crushing the separatists and putting an end to the 25-year-old war.
The United Nations estimated more than 4,500 civilians have been killed in the past three months.
The rebels said more than 1,000 civilians died Monday in a government raid, while the government said it rescued thousands after they broke through a barrier built by the insurgents that protects their last stronghold.
Human rights groups say the Liberation Tigers of Tamil Eelam are holding many people in the enclave against their will and using them as human shields. Those groups have also accused the government of indiscriminate shelling in the region. Both sides deny the allegations.
Thousands of civilians also fled in packed small boats, and they were picked up by navy patrols and transported to camps where Tamils who have escaped the war are being held. More than 2,000 people in about 100 boats were picked up Monday.
The Red Cross said about 50,000 civilians were still stranded, while Human Rights Watch put the number between 50,000 and 100,000.
A worker for Doctors Without Borders said hundreds of wounded were arriving at her hospital in Vavuniya, south of the war zone, in government-arranged buses, and some had died en route. The hospital is overcrowded with 1,200 people being accommodated in a facility with only 400 beds, said mental health officer Karen Stewart, according to a statement from the aid group.
U.N. high commissioner for refugees Antonio Guterres was concerned about the "dramatic situation" for civilians still in the war zone, said commission spokesman Ron Redmond.
"There are innocent civilians -- women and children -- caught in the middle of the conflict ... so the high commissioner is saying there should be a pause in the hostilities and the LTTE should allow civilians to leave," Redmond said.
Military spokesman Brig. Udaya Nanayakkara denied that 1,000 civilians died, saying 17 civilians were killed Monday by rebel shelling and by three suicide bombers.
"Our troops are rescuing the trapped civilians. It's the LTTE which is preventing civilians from fleeing," Nanayakkara said.
It was impossible to get independent accounts of casualties because journalists are restricted from the war zone.
The number of fleeing civilians made it clear that the government had vastly underestimated how many people were caught in the fighting.
"Both sides need to show far greater concern for civilians, or many more civilians will die," said Brad Adams, Asia director for the New York-based group Human Rights Watch.
A final government offensive "could lead to a dramatic increase in the number of civilian casualties," the International Committee of the Red Cross said.
"The situation is nothing short of catastrophic. Ongoing fighting has killed or wounded hundreds of civilians who have only minimal access to medical care," said Red Cross operations director Pierre Kraehenbuehl. "I cannot remember ... as much concentrated pain and exposure to violence with very, very minimal possibilities to reach anywhere that could be called safe."
The U.N. Children's Fund South Asia director Daniel Toole said he was worried about the safety of children still in the war zone because "our greatest fear is that the worst is yet to come."
The military spokesman said there was no fighting Tuesday evening, adding that it avoids using heavy, long-range weapons.
Nanayakkara said 39,081 civilians fled the war zone Monday, the most in a single day, and at least 13,000 people crossed over Tuesday, with the stream continuing.
State Department spokesman Robert Wood said the U.S. was pleased so many civilians had fled, but he remained concerned about what he called credible reports of increasing casualties and indiscriminate firing on civilians by rebel forces.
Encouraged by the exodus, the government Monday asked the rebels to surrender within 24 hours, warning of a final assault if they failed. The ultimatum expired at noon (2:30 a.m. EDT) Tuesday without a response from the rebels.
President Mahinda Rajapaksa rejected a call by British Prime Minister Gordon Brown for a pause in the fighting, his office said Tuesday.
The president's office said in a statement that Rajapaksa deemed a pause "unnecessary" considering the "unexpected exodus of civilians" when the two leaders had spoken by phone Monday.
Red Cross spokeswoman Sarasi Wijeratne said the organization could not confirm or deny the figures quoted by the military. According to figures received by the Red Cross, 11,000 people crossed the front lines Monday and another 5,000 came Tuesday, she said.
The U.N. refugee agency said it is ready to provide shelter and aid to tens of thousands of displaced civilians.
Redmond said the UNHCR was working with the government to make more land available for displacement camps because existing sites were overcrowded, he said.
The rebels have fought since 1983 for an independent state for Sri Lanka's ethnic minority Tamils. More than 70,000 people have been killed in the years of violence.
Associated Press writers Bharatha Mallawarachi in Colombo and Eliane Engeler in Geneva contributed to this report.