The Red Cross on Thursday sent a ferry to the battle zone in an effort to deliver food aid and to evacuate the wounded, but it could not reach the coastline due to the continuing violence.
"As fighting goes on unabated, civilians are forced to seek protection in hand-dug bunkers, making it even more difficult to fetch scarce drinking water and food," the ICRC said.
At least 2,700 civilians waded across a lagoon on Thursday while under fire from fighters from the Liberation Tigers of Tamil Eelam (LTTE), according to the military.
Four civilians were killed and 14 hurt as they attempted to cross the lagoon, which is currently a barrier between army forces and the LTTE, the military said.
At least another 1,000 were waiting to cross as night fell, although darkness had made it difficult to tell, said Brigadier Udaya Nanayakkara, a spokesman for the military.
It is impossible to verify the military's claims as it has blocked independent access to the conflict zone.
The LTTE has denied accusations by the military that civilians inside the war zone are being held as human shields and that they have opened fire on those trying to flee.
US presses IMF
Also on Thursday, Hillary Clinton, the US secretary of state, said the International Monetary Fund should reconsider a $1.9bn loan for Sri Lanka.
"We think it is not an appropriate time to consider that until there is a resolution of this conflict and that is what we are focused on," Clinton said in Washington.
Clinton's comments came a day after her president, Barack Obama, called on Sri Lanka's government to end "indiscriminate shelling" of civilian areas and for the Tamil Tigers to release civilians he said the group was holding in the area.
The Sri Lankan government has rejected calls for a ceasefire and the Tamil Tigers have refused to lay down their arms, despite being believed to be close to defeat in its 26-year battle for a separate Tamil homeland.
The UN estimates that about 50,000 civilians remain trapped in the narrow strip of land still held by Tamil Tiger forces.
Nearly 200,000 civilians have escaped the war zone in recent months and are being held in overwhelmed displacement camps.
Hundreds of civilians have been killed in the war zone in the past several days alone.