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The chairman of the ruling National Transitional Council (NTC), Mustafa Abdel Jalil, said its forces had called a two-day truce to allow civilians to leave as people streamed out of Sirte by the hundreds.
The truce order was issued on Friday (local time) but heavy rocket and mortar fire continued from both sides on Saturday, even though NTC commanders outside the town said they were trying to let civilians out.
NTC fighters in Sirte told Reuters that NATO planes had dropped flyers urging civilians to flee the fighting.
The prolonged battle for Sirte, encircled by anti-Gaddafi fighters and hit by regular NATO air strikes, has trapped people inside the town of about 100,000 through several fierce assaults over two weeks.
Fighting continued to the west and east of the town on Saturday. Loud thuds were heard coming from the town centre, and white smoke billowed into the sky while NATO planes roared overhead.
A truckload of supplies and a car carrying European ICRC workers were allowed to pass checkpoints manned by fighters loyal to the NTC.
The ICRC delivered medical kits for treating up to 200 people wounded in the fighting to Sirte hospital, as well as fuel to run the generator, a spokesman in Geneva told Reuters.
But the team of four aid workers, who also had security clearance from pro-Gaddafi forces, heard constant gunfire and so were not able to go into the hospital, the spokesman said.
"They went to the hospital but were not able to see patients, they didn't go into the wards," Marcal Izard said.
Doctors at the hospital -- which had no power -- told the aid workers there were 200 patients inside.
Izard said the ICRC team saw that the water tower of the hospital had been hit and damaged during the fighting.
Fighting remained heavy at a roundabout in the east of the town where NTC fighters have been held at bay for six days by artillery and sniper fire from pro-Gaddafi forces.
NTC commanders said the snipers were the main factor frustrating their advance. Reuters journalists have seen some anti-Gaddafi fighters run from the front under fire.
On Saturday, when a truck careened back from the roundabout carrying a dead NTC fighter, his comrades fired into the air and began to shout, "Muammar, the rat! He is killing us!"
Gaddafi loyalists and some civilians blamed NATO air strikes and shelling by NTC forces for killing civilians.
NATO and the NTC deny that. They and some civilians coming from the town say pro-Gaddafi fighters are executing people they believe to be NTC sympathisers.
The NTC is under pressure to strike a balance between a prolonged fight that would delay its efforts to govern and a quick victory which, if too bloody, could worsen divisions and embarrass the fledgling government and its foreign backers.
NTC chairman Abdel Jalil on Saturday acknowledged that disorganisation among its forces was hampering its efforts to take Sirte and Bani Walid, the other main holdout town.
"But the NTC is going on firmly regarding plans to liberate Sirte and Bani Walid," he said.
NTC officials have been caught off guard by the intensity of the resistance from the pro-Gaddafi fighters at Sirte and Bani Walid.
"This war is going to go on for a long time. Do you know why? It's because of the snipers. What will finish it is the rockets but they can't do that because of the civilians," a man called Mohammed said as he fled with his parents.
On Friday evening, one person was killed and six were wounded when pro-Gaddafi fighters emerged from Bani Walid and staged a surprise attack on the eastern flank of NTC forces stationed to the north, residents of the area told Reuters on Saturday.
The residents said it was believed to be the third or fourth time that such an assault had been attempted by the town's defenders since the start of the NTC's attack in early September.
As concern mounts for people in both towns, several residents have told Reuters they are leaving Sirte because they have not eaten for days.
"I am not scared. I am hungry," said Ghazi Abdul-Wahab, a Syrian who has lived in the town for 40 years.
Some residents say they had paid up to $800 for fuel to leave the city because it was scarce. Others said pasta and flour were now changing hands for large sums of money.
Doctors at a field hospital said an elderly woman died from malnutrition and they had seen other, similar cases.