Aid is finally flowing into Falluja, following the heavy US-led offensive that began nearly three weeks ago to wrest the city from rebel control.
The US prevented aid entering the city for weeks
The Iraqi Red Crescent told the BBC it was delivering aid on a daily basis.
But a spokesman says it is feared more than 6,000 people could have died in the assault and thousands of families are in critical need of assistance.
Meanwhile, Iraqi and US politicians have insisted elections will go ahead in January despite a plea for a delay.
A spokesman for Iraq's interim Prime Minister Iyad Allawi's said the government was determined to hold the elections on 30 January despite calls by some political parties for a six-month delay because of deteriorating security.
US ambassador to Iraq John Negroponte also said the poll must go ahead on time.
Convoys carrying food, water, medicine and blankets are moving around Falluja but there is still no running water or electricity.
According to the Red Crescent, 60 people came out to get assistance in one street alone.
The organisation's president, Dr Said Haqi, said it had now set up an office close to the city centre.
He described how one man in his mid-50s had approached them after staying in his house for the past month - apparently living on water and sugar.
In comments reported by the UN information network Irin, spokesman Muhammad al-Nuri said the Red Crescent believed more than 6,000 people may have died in the fight for Falluja.
He said it was difficult to move around the city due to the number of dead bodies.
"Bodies can be seen everywhere and people were crying when receiving the food parcels. It is very sad, it is a human disaster," Mr Nuri reportedly said.
No outbreaks of disease have been reported but the destruction is widespread, with at least a third of houses needing rebuilding, reports say.
It will be up to two months before about a quarter of a million civilians who fled the city will be able to return, Dr Said said.
US and Iraqi forces began their ground invasion of Falluja just under three weeks ago, but massive aerial bombardments of the city took place for weeks before that.
Troops in Falluja continue to encounter sporadic resistance.
Members of Iraq's electoral commission are said to be considering a petition from at least 15 mainly Sunni political parties to postpone general elections on 30 January for six months.
The parties cited the violence in Iraq and administrative problems as their reason for seeking the delay - adding that the elections needed to have political legitimacy.
But the US and Iraqi interim government have insisted the elections must go ahead.
There appears to be confusion over whether the commission, UN or interim Iraqi government would have the authority to delay the election, which is constitutionally bound to be held by the end of January.