AP Television News video from the Nahr el-Bared refugee camp showed women clutching children and piling up in pickup trucks, some waving white flags, as they tried to leave the partially destroyed camp.
Others fled on foot, and ambulances could be seen evacuating the wounded.
U.N. relief officials in another camp located a few miles to the south of Tripoli said they expected 10,000 Palestinian refugees from Nahr el-Bared to arrive through the night.
Refugees from Nahr el-Bared were seen raising white towels from windows and even waving white plastic bags. Boys carried babies, and a young boy and a woman helped an elderly woman, hurriedly walking on the side of the road as cars sped past carrying more refugees.
Many of the packed cars driving out had their windows blasted from the fighting.
Earlier, a U.N. convoy carrying relief supplies was hit during the fighting between the Lebanese troops and the Fatah Islam fighters.
The Lebanese army initially stopped the convoy of six trucks from the U.N. Relief and Works Agency, which aids Palestinians, from entering the camp. The convoy was later allowed in during a brief cease-fire.
But two pickup trucks and a water tanker got caught between the lines of the two sides and were hit as they entered the camp, said an UNRWA official, speaking to The Associated Press by telephone from the entrance of the camp.
The official said the convoy was then shot at as it tried to deliver aid. A car from inside the camp was also hit by fire in the area at the same time, said the official, who spoke on condition of anonymity because he was not authorized to talk to the media.
Al-Arabiya satellite television reported four civilians were killed in the incident. The UNRWA official said there were 15 civilian casualties but did not give a breakdown of dead and wounded. There was no official confirmation of either report.
The UNRWA official said earlier that he had reports dozens of buildings in the camp had been destroyed with residents trapped in the rubble.
At least 29 soldiers and 20 militants have been killed in the fighting since Sunday. The number of civilian casualties is not known, however, because relief workers, Lebanese authorities and journalists have had limited or no access to the camp.
Overnight, the Lebanese government ordered the army to finish off the militants who have set up in Nahr el-Bared, where 31,000 Palestinian refugees live on the outskirts of the northern port of Tripoli.
Black smoke billowed from the area Tuesday amid artillery and machine gun exchanges between troops and militants. Lebanese troops skirmished with Fatah Islam fighters, trying to seize militant positions on the outskirts of the camp.
"There are dead and wounded on the road, inside the camp," screamed a Lebanese woman, Amina Alameddine, who ran weeping from her home on the edge of the camp. She fled with her daughter and four other relatives after Fatah Islam fighters started shooting at the army from the roof of her house.
At the same time, Lebanese troops sought to flush out fighters hiding in Tripoli. Soldiers raided a building where Fatah Islam militants were believed to be hiding out, blasting an apartment with grenades, gunfire and tear gas.
They found no one in the apartment. As they pursued a militant hours later, he blew himself up by detonating an explosives belt rather than surrendering. None of the troops was injured.
Dozens of refugees angered by the assault on Nahr el-Bared burned tires in protest in the southern camp of Ein el-Hilweh, Lebanon's largest Palestinian camp. Protesters also burned tires in Rashidiyeh camp, farther south.
The Lebanese Broadcasting Corp. and camp residents reported that 200 Palestinians in Nahr el-Bared demonstrated against Fatah Islam, asking them to leave the camp.
Reports emerged from the camp of heavy destruction from the three days of bombardment by Lebanese artillery and tanks and militants who returned fire with mortars and automatic weapons.
"The shelling is heavy, not only on our positions, but also on children and women. Destruction is all over," Fatah Islam spokesman Abu Salim Taha told The Associated Press by telephone from inside the camp.