CAMP TARIQ, Iraq -- Iraqi forces battling their way into Fallujah repelled a four-hour counterattack by the Islamic State of Iraq and Syria (ISIS) on Tuesday, a day after pushing closer to the southern part of the militant-held city with the help of U.S.-led airstrikes.

A leading aid group meanwhile raised alarm over the unfolding "human catastrophe" in Fallujah, and renewed calls on warring parties to open up safe corridors for civilians to flee.

U.S. military officials in Baghdad tell CBS News that Iraqi forces have yet to advance into the city itself - and that won't be easy. Although a few hundred residents have managed to escape, tens of thousands more are thought to be trapped in the city, held hostage by ISIS militants intent on using them as human shields, CBS News correspondent Charlie D'Agata reports.

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The dawn attack unfolded in Fallujah's Nuaimiya area, most of which was captured by Iraqi troops the previous day, two special forces officers told The Associated Press. They spoke anonymously because they were not authorized to release the information.

ISIS militants used tunnels and snipers, and targeted Iraqi forces with six explosives-laden cars that were destroyed before reaching their targets, the officers said. Iraqi forces suffered casualties, but the officers didn't provide details.

Monday's push into Nuaimiya, a sprawling agricultural area, was the first attempt by Iraqi forces to enter the city, which fell to ISIS in January 2014. In recent days Iraqi forces had focused on driving the militants from outlying areas to tighten a siege on the city.

The clashes subsided by Tuesday afternoon, but the officers said further progress was slowed by roadside bombs planted by militants. The troops also paused to destroy tunnels in the area. The officers said 106 militants have been killed since Monday.

Fallujah was the first Iraqi city to fall to ISIS and is the last major urban area controlled by the extremist group in western Iraq. The militants still control the country's second largest city, Mosul, in the north.

The U.S.-led coalition and Iranian-backed Shiite paramilitary forces are helping the Iraqi army in the battle to retake Fallujah. But the fight is expected to be protracted, as the militants have had more than two years to dig in. Tens of thousands of civilians remain trapped in Fallujah and hidden bombs are believed to be strewn throughout the city.

"A human catastrophe is unfolding in Fallujah," Jan Egeland, the head of the Norwegian Refugee Council said, adding that only one family managed to escape from the town on Monday. Since the offensive began a week ago, 554 other families have escaped from areas surrounding Fallujah, which lies 40 miles west of Baghdad.

"Warring parties must guarantee civilians safe exit now, before it's too late and more lives are lost," Egeland added. The NRC said a lack of food, medicine, safe drinking water and electricity in the city is "pushing families to the brink of desperation."

In a weekly briefing in Geneva, the spokesman of the U.N. refugee agency, William Spindler, said 624 families -- or around 3,700 individuals -- have fled over the past week, citing figures by Iraqi authorities.

Spindler said the UNHCR "understands another 500 men and boys over 12 years old are held for security screening," which can take five to seven days. "We understand some 27 men were released" on Monday, he said.

The world's largest body of Muslim-majority countries expressed "deep concern" for the safety of civilians in Fallujah. In a statement, the head of the 56-nation Organization of Islamic Cooperation expressed support for the Iraqi government and the group's opposition to sectarian violence.

Some Sunni lawmakers in Iraq have accused the security forces of using indiscriminate force and say Shiite militias have committed abuses against civilians in mainly Sunni towns and cities. The government-sanctioned umbrella group of mostly Shiite militias is not taking part in the current push into Fallujah, according to officials.