HIT, Iraq (AP) — Iraq's counterterrorism forces are increasingly relying on air power in the battle to take full control of a key town in western Anbar desert from Islamic State militants, the elite force's commander said Thursday.

The fight for Hit, a small Euphrates River town west of the capital of Baghdad, has become pivotal in Iraq's campaign to claw back territory from IS in the sprawling province. The town sits along an IS supply line that links Iraqi territory controlled by the extremist group with its base in Syria.

Iraqi military officials and the U.S.-led coalition fighting IS believe that by clearing the town, the Iraqis can build on recent territorial gains in the vast desert of Anbar.

Gen. Abdel Ghani al-Asadi, who leads the counterterrorism troops, said at this point his men control most of Hit but that IS fighters are still hunkered down in far-flung areas of the town from where they are putting up heavy resistance.

He said that late into Wednesday night, front line commanders were calling in coordinates to coalition forces at the nearby Taqaddam air base, about 50 kilometers (31 miles) away .

Al-Asadi said his men are facing tough counterattacks because, unlike Ramadi — a far larger city and the provincial capital of Anbar — he believes Hit was a more important stronghold for IS.

In chatter that the commander overheard from intercepted IS radio communication, IS fighters were saying that "this is our headquarters and we will never leave this area," said Gen. al-Asadi.

On Thursday, Iraqi forces on the southern edge of Hit moved down the street of residential blocks, calling for coalition air support before advancing just tens of meters (yards) at a time. A handful of Humvees provided cover as a bulldozer was erecting a roadblock from rubble and abandoned vehicles to prevent potential suicide car bombings.

Meanwhile, dozens of Hit civilians holding white flags of peace were seen walking out of the town through rubble and across the front lines.

Iraqi officials estimate that more than 20,000 civilians have fled Hit since the launch of the operation to retake the town. Thousands more remain trapped inside the town and in its outskirts, al-Asadi said, slowing the last stages of the fight.

After overrunning Iraq's second largest city of Mosul in 2014, Islamic State militants swept across northern and western Iraq, linking the group's territories in Iraq and Syria and declaring their extremist caliphate. Despite a string of territorial victories against IS, the group still controls large swaths of territory straddling the Syria-Iraq border.

Associated Press Writer Khalid Mohammed in Hit, Iraq contributed to this report.