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Ammon News - AMMONNEWS - About 2,300 families have fled the Iraqi city of Fallujah over the past 24 hours, according to an international aid group, as government forces and their allies push forward with an offensive to retake the city from the Islamic State of Iraq and Levant (ISIL) group.

The Iraqi army said on Saturday it had gained control of Fallujah's main hospital, a day after recapturing the government compound in the centre of the city.

Fighters belonging to ISIL, also known as ISIS, still hold roughly 20 percent of the city and are entrenched on its northern districts.

Humanitarian agencies working on the outskirts of Fallujah, located 50km west of the capital Baghdad, said they were struggling to cope with the heavy flow of displaced civilians fleeing the violence as the offensive continues, pressing on towards ISIL-held Mosul .

"In the last 24 hours, more than 2,300 families have actually managed to leave Fallujah, and to be honest, there's very little space for them in Amariyat al-Fallujah, Habbaniyah Tourit City and Khaldiyah, which is where they are escaping to," Nasr Muflahi, Iraq country director for the Norwegian Refugee Council in Erbil, told Al Jazeera on Saturday.

More than 20,000 people have fled Fallujah in the past two days alone, according to the UN's refugee agency (UNHCR).

"We are now finding it difficult to cope with the numbers that are coming out of Fallujah, especially in terms of delivering safe drinking water. We're down to the bare minimum of three litres per person, and we're not really sure how long we can continue to do that," said Muflahi.

The UN and the Iraqi government have set up camps for 60,000 displaced civilians in Anbar province, but have warned there is little capacity to absorb any more people.

New arrivals, many of whom have been trapped by fighting for weeks, reach relative safety to find overcrowded camps and settlements.

"The people coming out of Fallujah are telling us horrific stories of how they were treated - the lack of food, no services, no electricity," said Muflahi.

"We are doing our utmost, with other agencies, to respond to their needs, which is shelter, water and food," added Muflahi. "These are things that we need to prioritise, and we need to now rather than later."

Some 70,000 people are estimated to have fled Fallujah, and another 60,000 are expected to leave over the next several days, according to the UNHCR . The agency estimates that up to 150,000 displaced people may soon be in need of urgent humanitarian assistance.

"Thousands of families may also remain trapped in Fallujah," Stephane Dujarric, spokesman for the UN secretary-general, told reporters on Saturday. "These are estimates ... however, we remain very concerned about the safety and the wellbeing of the people still in Fallujah."

Rights groups have voiced concern over reports of abuse, mistreatment and extrajudicial killings. And Sunni politicians have called on Iraqi Haider al-Abadi, Iraqi prime minister, to investigate the rising number of alleged abuses.

READ MORE: 'ISIL fighters sneaking out of Fallujah with civilians'

Iraqi security forces fear that ISIL fighters may be hiding among the displaced.

Shia units known as the Popular Mobilisation Forces fighting alongside government troops have been separating males from their families, detaining the men to put them through a security screening process.

"The process may take a few days, it may take a few months," said Muflahi. We can't confirm any extrajudicial killings."

The government lost control of Fallujah in 2014, months before ISIL took Iraq's second largest city, Mosul, and swept across large parts of the country.

As a result of escalating violence over the past two years, more than 3.4 million people are now displaced across Iraq - more than half of them children.

*Al Jazeera