Fierce fighting has followed the coalition troops' advance through Fallujah's southern suburb of Nuaimiya.
The Iraqi government push to regain control of the city is expected to be one of the biggest battles fought against IS militants so far.
As the fighting intensifies, so do concerns for civilians trapped in city.
United Nations Refugee Agency spokesman William Spindler says there have already been many casualties.
The UNHCR has also received reports that hundreds of people are being used by IS as human shields in the centre of Fallujah.
"UNHCR has received reports of casualties among civilians in the city centre of Fallujah due to heavy shelling, including seven members of one family on the 28th of May. There are also reports of several hundred families being used as human shields by ISIL in the centre of Fallujah."
Iraqi authorities say 3,700 people have fled the city over the past week.
Forces have been helping transport families escaping the city, but about 50,000 are still believed to be trapped.
UN Assistance Mission for Iraq spokeswoman Lise Grande fears it will be a long time before those people receive desperately needed food and clean water.
"We understand that there are widespread food shortages and there have been since January when access to Fallujah became extremely difficult for commercial convoys. I think what we would characterise this as mass food deprivation. The shortages have resulted in literally thousands of families no longer able to eat three meals a day. Some of the families that have managed to escape have said 'look we cut down to one meal a day in February, by March we were having to skip meals every two or three days and sometimes in the course of the week we are only eating on one to at most three days of the week."
Two citizens of Fallujah who managed to get out, Layla and Allahen, say life under the control of IS has left many traumatised.
"We lived in panic. For bread - we couldn't make any for three months. We ate dates and yoghurt. We were able to eat that, thank God. We couldn't leave. We had no money."
"For two and a half years in my area, families started to suffer when IS closed exit routes from the city. They started to suffer from psychological problems, some families committed suicide, some of them set fire to themselves, and some of them drowned their children. Some families threw their kids on to the streets and left them their on their own because they didn't have any food for them. As God as I witness, everything I say is true. This happened."
Becky Bakr Abdullah is from the aid agency Norwegian Refugee Council.
The group helps run humanitarian camps in the region.
Ms Bakr Abdullah says all parties need to make the safety of civilians a priority.
"Around 554 families have managed to escape the outskirts of Fallujah where there is also intense fighting, but unfortunately from inside Fallujah town we only have reports of one family that managed to escape. The fighting is intensifying hour by hour. Fallujah town is now being bombed by three sides, south, west and east, they're now bombing the town. We can only hope and ask all involved parties to this conflict to grant the 50,000 individuals safe exits out."
But it's unlikely IS would allow civilians to leave Fallujah.
An Iraqi military and strategic analyst, Amir al-Saadi, says IS militants have been preparing for an assault by Iraqi forces since they first seized the city in 2014.
"IS has wreaked huge devastation and havoc in the city of Fallujah. The IS militants haven't left any area of the city alone, but have planted booby traps and mines on it. I have heard that directly from families who have survived. The extremists are making booby-traps in houses and neighbourhoods in order to prevent families from getting out, to use them as human shields. They aim to destroy the infrastructure of the city completely. The militants have dug parallel trenches along the wings of the armed forces' advance.''
The Iraqi military has urged the people remaining in Fallujah to either leave the city or stay indoors - a life-threatening dilemma for civilians either way.
