An AFP reporter at a mosque field hospital west of the city said hundreds of Sirte residents were fleeing in packed vehicles, with some people even sitting on top of possessions piled high in the rear of pick-ups.
"There are so many rockets now. Yesterday there were a lot of attacks. We just could not stay any longer," Ali Faraj said as a National Transitional Council fighter checked his identity and those of women and children crammed in his car.
A Red Cross team, which delivered desperately needed supplies to medics in the besieged coastal city on Saturday, said the hospital had come under rocket fire as new regime forces stepped up their assault on Gaddafi diehards.
A large force of NTC fighters pushed in from the south to lay siege to the Ouagadougou Conference Centre, a showpiece venue close to the Ibn Sina hospital where Gaddafi hosted the launch of the African Union.
Intense exchanges raged for at least two hours despite pleas from the Red Crescent for a lull while the International Committee of the Red Cross team made its delivery, NTC fighters said.
"It's a dire situation," ICRC team leader Hichem Khadhraoui told AFP. Staff at the Ibn Sina hospital told the team that "because of lack of oxygen and fuel for the generator, people are dying."
Other wounded or ill people cannot get to the hospital because of the fighting and NATO air strikes, Khadhraoui added. "Several rockets landed within the hospital buildings while we were there. We saw a lot of indiscriminate fire. I don't know where it was coming from," he said.
After the ICRC team went in, NTC fighters launched a ferocious attack with rockets, anti-tank cannons and machinegun fire from a position less than a kilometre (half a mile) from the hospital. Gaddafi loyalists responded with mortar and sniper fire.
"We were surprised" that the attack took place while the team was visiting, Khadhraoui said, adding that they had "contacted all parties to say we were going in."
The hospital's water tower was hit, forcing staff to bring in water from outside, he said.
East of Sirte on Sunday, NTC fighters returning from the front said they had captured a neighbourhood in the southwest of the city which was home to many Gaddafi clansmen." Ninety-five per cent of Buhadi is under our control," fighter Drisi Mayar said.
"This was a stronghold of Gaddafi. A lot of his relatives and clan members lived there. There was a small military base. We took control yesterday. We had small clashes but it is under our control."
Hundreds of vehicles also streamed out of Sirte on the eastern front during a lull in fighting on Sunday.
One man leaving with his family who gave his name only as Muftah said: "The situation is absolutely pathetic, especially in the hospitals. We have no oxygen, no medicines. Wounded people die even before reaching the hospital.
"Many people have broken open the pharmacies in the city to bring medicines to the hospital but even that is exhausted now."
The ICRC had been trying for weeks to enter Sirte, which has been under siege by NTC forces since the middle of last month.
It had sought to bring medical supplies in by boat but abandoned that idea because of security concerns. Sirte's port is now in the hands of NTC forces and there has been fierce fighting around it since its capture.
Khadhraoui's team on Saturday included a doctor, a first aid medic and a logistician, he said. It delivered about 150 body bags and 300 "war wounded kits" consisting of drips, drugs, gauze and other medical equipment.
The team did not tour the wards or carry out a full assessment of the hospital's medical needs but hopes to return soon to bring in more supplies. "Oxygen is the main thing they asked for," Khadhraoui said.
Some of the hundreds of residents fleeing Sirte said there had been civilian casualties when residential buildings were hit, either by artillery fire from besieging new regime forces or by NATO air strikes.
"I left with my family as we are caught between NATO bombings and shelling by rebels. NATO, in particular, is bombing at random and is often hitting civilian buildings," a man who gave only his first name, Ali, said.
A rocket killed two children on Saturday when their family joined the desperate exodus.
The two children killed "were torn to pieces," said Ahmed Abu Aid, a field medic on the western side of Sirte. "They collected the body parts in bags."
Redwan Abdulrahim, his small truck piled high with suitcases and other possessions as he drove out on the coast road to the west, said the situation in Sirte had become intolerable.
"It was really bad. We didn't know where the bullets and rockets were coming from."