The Post’s Hugh Naylor reports from the front lines of the fight to remove the Islamic State from the Iraqi city of Fallujah. The assault is being led by the military, police and elite counterterrorism forces, but powerful Shiite militias are eager to be involved, as well. (Hugh Naylor/The Washington Post)

Only a few days ago, the Islamic State ruled this verdant Iraqi farmland. By Saturday, the area was firmly controlled by powerful Shiite militias, which delivered swift blows to the Sunni extremists and severed their supply lines to nearby Fallujah.

Iraq’s government has ordered the militiamen to stay away from the battle to drive the Islamic State out of Fallujah, fearing further sectarian unrest from their presence inside the Sunni stronghold.

The assault on the city — a key test in the U.S.-supported campaign to oust the Islamic State from Iraq — is being led by the military, police and elite counterterrorism forces. But their progress has slowed, and Shiite militia leaders on the outskirts here, such as Hadi al-Amiri, appear antsy.

He wants to send in the Iranian-backed fighters of his Badr Organization, which he commands.

“No one can stop us from going there,” Amiri said Saturday at a commandeered farmhouse about a mile west of Fallujah.

Such a move into Fallujah could cause serious problems. In early 2014, the Islamic State easily took control of the city by exploiting the anger of its residents against the Shiite-dominated government in Baghdad.

And Sunni leaders from the area, in Iraq’s western Anbar province, have expressed extreme discomfort with the presence of the militias. Human rights groups have accused Shiite militiamen of brutal treatment of Sunnis suspected of ties to the Islamic State.

Iraqi Prime Minister Haidar al-Abadi has vowed to address Sunni grievances, which include allegations of neglect and discrimination by authorities. And the Iraqi military has ordered most of the militias — formally called the popular mobilization units — to keep out of Fallujah. The units are mostly Shiite groups, some of them backed by Iran, but they also include smaller outfits of Sunnis, Christians and other religions.

But soldiers and police say they have struggled to fight their way into Fallujah. The Islamic State has used the estimated 50,000 or more civilians still trapped there in dire conditions as human shields, and a sudden advance could inflict many casualties.

“In this battle, Daesh has really focused on using human shields as a tactic because we have adapted to their other tactics,” said Maj. Gen. Thamir Ismael, the area commander for the SWAT police force. Daesh is the Arabic acronym for the Islamic State, also known as ISIL and ISIS.

Amiri’s men have momentum, though, and they appear eager to join the assault on the city. In an interview, the 61-year-old Amiri gave a 10-day deadline for civilians to leave Fallujah. Then the militiamen, along with pro-government tribal militants, federal police and soldiers, would prepare to storm it, he said.

“Right now, the only thing that is stopping us from going there are civilians,” he said.

Wearing a hat and khaki pants, he directed operations against the few pockets of Islamic State fighters still left in the Fallujah suburb of Saqlawiyah. His fighters and Iraqi police watched in deference as he ordered a bulldozer to drive ahead of fighters to protect them from possible Islamic State suicide bombers.

“Hurry up so we can finish our job here,” he said over a two-way radio.

As he spoke, the thuds of exploding artillery and the cackle of machine-gun fire could be heard in the background. Militiamen, unfazed by the sounds of war around them, took smoke breaks and feasted on rice and lamb.

In nearby fields, the fighters fired mortars and Katyusha rockets at Saqlawiyah, where an unknown number of civilians are thought to be residing. Dozens of families fled the area on Saturday. Officials in the area planned to house them in a nearby camp for displaced people.

Amiri’s tough statements underscore the strength that the militias have in Iraq. And Amiri has become one of the most powerful men in that country.

After the military collapsed during the Islamic State offensive in June 2014, the militias stepped in, blunting the extremist group’s advance. As a result, many Iraqis view them as heroes.

But their rising power has alarmed many others in Iraq and officials in the United States, who fear that they have undermined Abadi’s authority. U.S. coalition aircraft refuse to provide air cover for militias such as Amiri’s. They limit airstrikes to operations that involve Iraqi soldiers and counterterror forces.

At the moment, Amiri’s fighters appear to have little need for U.S air cover. Scores of vehicles rumbled down dirt roads that the Badr fighters and other militiamen cleared in recent days, sometimes using hand-to-hand combat.

Despite facing intense resistance, including snipers and suicide attacks, they rapidly drove Islamic State militants out of areas west of Fallujah. The attack cut the city off from Saqlawiyah, depriving the Islamic State of supplies of food, water and weapons.

“They used tunnels to mount surprise attacks on us,” said a media relations officer with the Badr Organization as he drove through what had until recently been a battlefield.

He pointed to burned-out tanks and several destroyed farmhouses, which Islamic State militants had rigged with explosives. The roofs of those buildings had neatly pancaked atop the rubble.

“It took time to clear these areas, but we have gained much experience and we are moving fast.”

Mustafa Salim contributed to this report.