The UN refugee agency (UNHCR) warned that food and medical shortages and sustained shelling could provoke a mass exodus.
An agency official in Iraq, Bruno Geddo, said: "The worst is yet to come."
Last month the Iraqi government launched an offensive to recapture west Mosul, which is the last major urban centre controlled by IS.
Fighting has intensified in recent days as the government forces have retaken a series of neighbourhoods from IS, but the militants still hold an estimated 40% of western Mosul.
The UN says 600,000 people are still in IS-controlled areas, including 400,000 trapped in siege-like conditions.
"They are desperate for food. They are panicked," said Mr Geddo, speaking from the Hammam al-Alil camp for people displaced by the fighting, a few miles south of Mosul.
"There is a shortage of fuel, of food, of electricity.
"People have resorted to burning furniture, old clothes, anything they can use to keep warm at night, because it is still raining heavily and the temperatures at night in particular drop significantly," he said.
He said many people were surviving on just bread and water, and eating once a day.
Medical charity Doctors Without Borders have reported treating children for "severe malnutrition" after they escaped the city.
The UN says many people are scared to leave their homes because of Islamic State snipers, but around 157,000 have reached a transit centre since the offensive began.
Between 8,000 and 12,000 people were now arriving at the transit centre each day, Mr Geddo said.
He said those who were making it through were telling staff at the centre they preferred "to take the risk of dying while I stand a chance to be free and safe again with my family".