(CNN) ISIS is targeting civilians attempting to leave the militant-held city of Falluja, as Iraqi forces and militia attempt to wrest back control of the city, a European non-profit operating in Iraq says.

As many as 50,000 residents remain trapped in the center of Falluja as Iraqi security forces close in, the Norwegian Refugee Council (NRC) told CNN Monday. Many of those who try to escape the clutches of ISIS are being targeted and shot by militants, it said.

"Our biggest fears are now tragically confirmed with civilians being directly targeted while trying to flee to safety," NRC Country Director in Iraq Nasr Muflahi said.

"This is the worst that we feared would happen to innocent men, women and children who have had to leave everything behind in order to save their lives."

Falluja, which lies 65 km (40 miles) from the capital Baghdad, has been held by the militant group since 2014, and is the subject of a concerted push to retake ISIS-controlled territory across Iraq and Syria.

Almost 3,000 families have safely escaped the city since late May, the NRC said. Most are being housed in displacement camps, and the NRC has warned of water shortages should the tens of thousands remaining in Falluja manage to escape.

Struggle to escape

Many of those now in displacement camps outside Baghdad say they barely made it out of the villages and towns surrounding the besieged city.

JUST WATCHED UNICEF: 20,000 children trapped inside Falluja Replay More Videos ... MUST WATCH UNICEF: 20,000 children trapped inside Falluja 01:40

One man CNN spoke to said ISIS fighters came to his home and told him he needed to go to the center of Falluja to serve as a human shield for the terror group.

"It was an order," Taleb Farhan, a resident of Karma on the outskirts of Falluja, said. "If you refused they'd shoot you on the spot."

One family hid in their home, but left the door ajar in an attempt to fool the militants.

"When ISIS came the thought the house was empty," Thamir Ali, also from the town of Karma, says. "Other families didn't do that. ISIS took them away or killed them in their homes."

Some who made it out told CNN that four families -- 32 people in total -- hid in the marshes for four days, drinking dirty, brackish water and eating old dates before they were finally able to escape.

They ended up in a dusty, displaced persons camp in Abu Ghraib outside of Baghdad, where there is very little in the way of food and other supplies.

They hope that when ISIS is defeated, they will have homes for them to return to.

CNN witnessed extensive damage to homes and other buildings -- very few structures remain standing near Falluja, after the area has been pounded by coalition and Iraqi air force strikes.

Ground gained

Iraqi security forces, with the support of other militia, stormed the first neighborhood in southern Falluja at the weekend after recapturing the district of al Nuaimiya -- 5 kilometers (3.1 miles) from central Falluja.

Separately, Iraqi security forces recaptured the key small town of Saqlawiya, 10 kilometers (6.2 miles) northwest of Falluja, on Saturday. Troops there "raised the national flag over the town's main buildings," Iraq's Joint Operation Command said.

The Iraqi military swept through the streets conducting searches for "hundreds of IEDs" that ISIS planted throughout Saqlawiya, military officials said.

Coalition warplanes carried out an airstrike on ISIS militants as they were trying to escape on a raft on the Euphrates River, south of the city, killing everyone aboard raft, the officials said.

The takeover of Saqlawiya comes a day after coalition warplanes bombed ISIS command centers and tunnel networks in Falluja, killing dozens of militants.

The airstrikes targeted ISIS tactical units, intelligence sources said.

Iraqi security forces said Saturday's advances mean ISIS is losing its last strategic foothold between Falluja and the rest of Anbar province to the west, as well as other areas to the north.

In January 2014, Falluja became the first Iraqi city to fall to ISIS -- dealing a blow to Iraq as well as Western countries battling the terrorist group. Saturday is the first time more than two years Iraqi security forces have come this close to a full takeover, army officials said.

Parallel offensive

Meanwhile, Syrian forces made progress against the militant group in the region surrounding the province of Raqqa in the first push into the area since 2014.

The Syrian Army, joined by Russian forces, advanced into the province on Saturday "in a major offensive backed by Russian warplanes," according to the London-based Syrian Observatory for Human Rights (SOHR).

Aid groups say the military advance to enter Raqqa, located on the north bank of the Euphrates River, comes after three days of intense fighting.

"At least 26 ISIS militants and 9 Syrian military soldiers were killed," said SOHR.

This is the first time in two years the Syrian Army has been inside Raqqa province.

As Syrian forces fight on the ground, they continue to receive help from the international coalition targeting ISIS from the sky.

U.S. officials say airstrikes are having a major effect as Syrian and Iraqi troops and their allies continue to claw back ground from ISIS in the two countries.

Jets take off every few minutes from the deck of the USS Harry Truman, which has about 75 strike aircraft on board. The aircraft carrier recently moved to the Mediterranean to be closer to targets in northern Syria, increasing the intensity of the bombing runs.

To date, the coalition has made over 4,000 airstrikes in Syria, and over double that number in Iraq.

A humanitarian crisis

The civil war in Syria has claimed the lives of more than 250,000 people nationwide and displaced more than 10 million , according to the United Nations.

As the battle for Falluja intensifies, an estimated 50,000 people, including 20,000 children , are trapped between opposing fighters, according to the United Nations.

Men and boys who refuse to fight for ISIS are being killed, and civilians have died in heavy shelling, the United Nations says.

Aid groups say ISIS is using thousands of civilians there as human shields.