(CNN) Suad had feared for so long that she would die. A mother in Falluja, Iraq, her city has been at the center of war for more than a dozen years and now it is the epicenter of a battle between ISIS and Iraqi and Kurdish forces backed by the United States-led coalition.

The coalition's push to take Falluja from ISIS, which began late Sunday, prompted ISIS gunmen to go door to door in the city and yank men, their wives and children from their homes.

ISIS "kept moving us from one damaged, deserted house to another," Suad said. She, her husband and her children ran as fast as they could to avoid being shot in the crossfire.

"We tried going back to our house, but when we saw lots of other families fleeing we joined them," she said. "I was carrying my 2-year-old daughter, Hana, as I ran barefoot to reach the other families. My husband was behind us all the time trying to catch up with us as he can't run. That's how we left, taking nothing with us, not even our money. I even forgot to take my mobile phone.

Hundreds of people like Suad and her family have fled the city and its surroundings since the Falluja offensive began last week. More than 500 families have arrived in displacement camps outside the city, the Norwegian Refugee Council said, including almost 300 since Sunday afternoon alone. All the recent arrivals are from Jumeila on the city's outskirts.

"We feel safe here now," Suad said. "My children are getting food, and my husband is getting treatment in (the) hospital. I can't think of anything beyond that. Arriving here safely was what mattered, and it's a big relief."

But danger remains. Water shortages in the camps are an increasing concern.

"Last night we've had the biggest number of displaced families reaching us so far," said Nasr Muflahi, the Council's country director in Iraq.

"Our resources in the camps are now very strained, and with many more expected to flee we might not be able to provide enough drinking water for everyone," Muflahi said. "The priority remains providing safe exits for at least 50,000 civilians believed to be trapped in the center of Falluja. We expect bigger waves of displacement the fiercer the fighting gets."

Storm the city

Iraqi forces started their operation to retake the ISIS stronghold with the help of Iraqi and coalition air support, Iraqi military spokesman Yahya Rasoul said on Iraqi state TV.

"With God's blessing we have launched the third phase of the operation to storm the center of Falluja city -- by our heroes in the counterterrorism forces, units of the Iraqi army and Anbar police," Rasoul said.

"There is heavy air cover for this operation from our heroes in the Iraqi air force and the coalition."

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Iraqi forces retook the village of Nuaimiya, just south of Falluja, closing in on the city itself, al-Iraqi TV reported.

Earlier Monday, Iraqi military units and supporting militia captured a handful of settlements from ISIS near Falluja, including the town of Saqlawiya, about 10 kilometers (6.5 miles) northwest of the city, and the villages of al-Buaziz, al-Bu Efan and al-Shiha, north and west of Falluja, state-run TV reported.

Iraqi government troops, backed by Shiite militias known as the Popular Mobilization Units and an air campaign by the U.S.-led anti-ISIS coalition, launched the offensive last week to retake the ISIS stronghold, about 65 kilometers (40 miles) west of Baghdad, first targeting outlying settlements.

Capturing Karma, about 16 kilometers (10 miles) northeast of Falluja on Thursday, brought most of the territory east of the city under government control.

More than 5,000 Americans are providing training, advice and assistance to Iraqi forces and flowing equipment to them, United States Col. Steve Warren, a spokesman for the coalition, said on CNN's "New Day" Monday.

Iraq was "invaded by these animals" of ISIS, he said, but the terror group's grip on territory is slipping in both Iraq and Syria, he said.

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Tens of thousands at risk

The U.N. refugee agency has warned that an estimated 50,000 people are at risk, caught in the city turned combat zone as the Iraqi military's bombardment has intensified.

"There are reports of a dramatic increase in the number of executions of men and older boys in Falluja (who are) refusing to fight on behalf of extremist forces," said Leila Jane Nassif, the U.N. agency's assistant representative in Iraq.

"And many people have been killed or buried alive under the rubble of their homes in the course of ongoing military operations."

Hundreds, mostly women and children, fled Falluja on Friday as Iraqi soldiers attacked to drive ISIS from the city, the Iraqi military said.

Security forces evacuated about 760 people who escaped from eastern and southeastern Falluja, the military said.

"Food is scarce in the city," said Um Ahmed, a 40-year-old living in Falluja with her family. "We have mostly been relying on dates for our meals."

The Norwegian Refugee Council reports the residents of Falluja have been faced with food shortages, lack of electricity and hospitals have depleted supplies of medical supplies.

"Civilians are trapped inside the city of Falluja as fighting intensifies. With every moment that passes, their need for safe exits becomes more critical. We are working around the clock in the displacement camps to provide water and emergency food rations to people who manage to flee the violence, but getting there in the first place is near impossible for those in the city center," said Nasr Muflahi, Norwegian Refugee Council country director in Iraq.

Photos: The battle for Falluja Iraqi government forces drive their armored vehicle through a street in Falluja, Iraq, on Monday, June 27, after retaking the city from ISIS. A senior Iraqi general announced on state TV that the battle for Falluja is over as Iraqi troops retook the final ISIS holdout in the city. Hide Caption 1 of 29 Photos: The battle for Falluja Supplies are lined up at a camp for displaced civilians on Saturday, June 25. Hide Caption 2 of 29 Photos: The battle for Falluja Iraqi security forces evacuate an injured soldier during fighting with ISIS militants on Wednesday, June 15. Hide Caption 3 of 29 Photos: The battle for Falluja Iraqis who have been displaced from their homes peer out of a tent at a camp outside Falluja on Tuesday, June 14. Hide Caption 4 of 29 Photos: The battle for Falluja Smoke rises after airstrikes by U.S.-led coalition warplanes on Sunday, June 12. Hide Caption 5 of 29 Photos: The battle for Falluja A boy sleeps in his mother's arms after fleeing from their home in Falluja on Monday, June 6. Hide Caption 6 of 29 Photos: The battle for Falluja Smoke rises on the front line of fighting between ISIS militants and pro-government Iraqi fighters on Sunday, June 5. Hide Caption 7 of 29 Photos: The battle for Falluja Pro-government forces and Shiite militia fighters fire rockets toward ISIS fighters in Falluja on Saturday, June 4. Hide Caption 8 of 29 Photos: The battle for Falluja Iraqi soldiers help civilians from a vehicle outside a camp near Falluja on Friday, June 3. Hide Caption 9 of 29 Photos: The battle for Falluja Internally displaced civilians flee their homes after crossing the Euphrates River outside Falluja on Thursday, June 2. Hide Caption 10 of 29 Photos: The battle for Falluja Iraqi counterterrorism forces face off with ISIS militants in Falluja's Nuaimiya neighborhood on Wednesday, June 1. Hide Caption 11 of 29 Photos: The battle for Falluja Blindfolded men get led to a bus after being detained during the fighting on June 1. Hide Caption 12 of 29 Photos: The battle for Falluja A member of pro-government forces fires at a front line in the Albu Huwa area south of Falluja on June 1. Hide Caption 13 of 29 Photos: The battle for Falluja Grenades and shells lie on the ground in Falluja during fighting between Iraqi forces and ISIS on June 1. Hide Caption 14 of 29 Photos: The battle for Falluja Smoke rises as Iraqi forces face off with ISIS militants on the southern edge of Falluja on Tuesday, May 31. Hide Caption 15 of 29 Photos: The battle for Falluja Iraqi forces gather ahead of an operation to retake Falluja on Sunday, May 29. Hide Caption 16 of 29 Photos: The battle for Falluja Pro-Iraqi government fighters carry ammunition in the village of Harariyat on Saturday, May 28. Hide Caption 17 of 29 Photos: The battle for Falluja A police officer places an Iraqi federal police flag next to a Shiite militia flag outside Falluja on May 28. Hide Caption 18 of 29 Photos: The battle for Falluja Iraqi government forces engage with ISIS fighters near the village of al-Sejar on Thursday, May 26. Hide Caption 19 of 29 Photos: The battle for Falluja A pro-government fighter rests after battling ISIS near al-Sejar on May 26. Hide Caption 20 of 29 Photos: The battle for Falluja Iraqi government forces advance near al-Sejar on May 26. Hide Caption 21 of 29 Photos: The battle for Falluja Iraqi security forces surround an ISIS drone that was shot down outside Falluja on May 26. Hide Caption 22 of 29 Photos: The battle for Falluja Iraqi government forces take position east of Falluja on Wednesday, May 25. Hide Caption 23 of 29 Photos: The battle for Falluja A pro-government fighter holds position in the village of al-Shahabi, east of Falluja, on Tuesday, May 24. Hide Caption 24 of 29 Photos: The battle for Falluja Iraqi government forces join an operation in al-Shahabi on May 24. Hide Caption 25 of 29 Photos: The battle for Falluja Pro-government forces advance toward Falluja on Monday, May 23. Hide Caption 26 of 29 Photos: The battle for Falluja A fighter with the Badr Brigades, a Shiite militia, takes an ISIS flag down outside Falluja on May 23. Hide Caption 27 of 29 Photos: The battle for Falluja A fighter from a Shiite militia takes a combat position outside Falluja on May 23. Hide Caption 28 of 29 Photos: The battle for Falluja Iraqi Shiite fighters fire artillery near Falluja on May 23. Hide Caption 29 of 29

Targeting civilians

Facing increasing pressure, ISIS has lashed out with attacks on civilians in Iraqi and Syrian cities.

Three blasts shook parts of the Iraqi capital Monday morning, according to Iraqi security sources. No group immediately claimed responsibility.

A bomb in a market in Baghdad's al-Shaab neighborhood killed seven people and wounded 20 others. A separate explosion in Sadr City killed another two and wounded 10.

A suicide car bomber tried to ram into a police station in al-Tarmiya but was stopped from reaching the target. He detonated the bomb, killing himself and causing material damage but no other injuries, sources said.

Multiple offensives

Farther north, thousands of Kurdish Peshmerga troops are involved in an offensive to retake formerly Kurdish villages near the ISIS-held town of Mosul, Kurdish officials say.

The Peshmerga-led ground offensive, backed by coalition air support, was launched early Sunday to recapture villages near Khazir, east of Mosul.

The operation includes about 5,500 Peshmerga fighters.

The move comes ahead of a joint offensive by Kurdish forces and Iraqi troops to take back Mosul, Kurdish media said.

Across the border in Syria, a U.S.-backed alliance of Kurdish and Arab forces last week pushed into territory north of Raqqa, the de facto capital of ISIS.

Sarmad al-Jilane, an activist with the monitoring group Raqqa is Being Slaughtered Silently, confirmed reports that ISIS has let some residents of that city flee to the surrounding countryside or Deir Ezzor as Kurdish and Arab forces pushed forward.