ISIS losing its grip on the Iraqi city of Fallujah

FALLUJAH -- The Islamic State of Iraq and Syria (ISIS) is losing its grip on the Iraqi city of Fallujah, but fighting continues.

It's a major test of whether the U.S. supported government can reclaim Iraq. ISIS held the city of 300,000 people for two years. U.S. Military advisers say about a third of Fallujah is secure.

Iraqi forces push into center of Fallujah

Iraqi gunners fired towards ISIS machine gun positions near the heart of the city.

The militants retaliated with mortars as a column of Iraqi army Humvees inched forward. As residents still trapped inside Fallujah scramble to get out.

Tank commander colonel Abdulla al Shamra told CBS News the ISIS tactic of hiding among civilians has slowed his progress.

It's hard to thread the needle with a tank.

"I can't kill the children, I can't kill the old women, you know all of them like my father, my grandfather, my sisters," he said.

Advancing soldiers face what one commander called a "spider's web" of homemade bombs, and nests of snipers.

One bullet tore through the leg of soldier Haider Moeen Abas, who seemed more annoyed that he had to stop fighting.

Others won't return.

While we were here an ambulance pulled up, there was a soldier inside, they didn't want us to film, what happened the soldier walked into a booby trapped house, a bomb went off and he didn't make it.

These Iraqi special forces just finished 17 days of intense urban warfare.

Refugees desperate for food and water outside Fallujah

What was the most difficult part of the fight?

"The street by street is very tough," a soldier said, "So we decided to fight like them and now we're beating them at their own game."

As the sun began to sink, CBS News found a group of hungry young soldiers settling down to their first meal all day.

Battle-hardened and weary, having driven ISIS out of two other cities, it's a brief respite as the fight in Fallujah drags on.

The U.S. Department of State pledged another $20 million in humanitarian aid on Tuesday, to help tens of thousands of residents who fled the city since the assault began last month, now in urgent need of food, water and shelter.