Updated

Iraqi commanders are preparing to dislodge Islamic State group fighters from pockets of territory in Fallujah's northern and western neighborhoods where the militants have dug in after largely fleeing their positions in the city center last week.

Before Iraqi forces rolled into central Fallujah under cover of U.S.-led coalition airstrikes, they were bogged down for weeks, trying to push through deep defensive trenches, tunnels and houses converted into bunkers by IS militants on the city's southern edge. Now looking to the city's north, Iraqi commanders expect to encounter a similarly fierce fight.

"It's not going to be easy," Iraqi special forces Brig. Gen. Ali Jameel said of the upcoming battle for the last pockets of IS resistance where an estimated 100 militants are largely surrounded.

"They are going to fight to the death because they have nowhere to run," he said.

Prime Minister Haider al-Abadi launched the offensive to retake Fallujah from the Islamic State group on May 22. The Sunni-led extremist group has held the city west of Baghdad for more than two years.

Last week, Iraqi forces raised their flag above a government complex in central Fallujah and declared victory, saying 80 percent of Fallujah was under their control. It was some much-needed positive news for al-Abadi amid growing anti-government protests and civil unrest in Baghdad.

But nearly a week later, Iraq appears to have only a fragile hold on the territory its forces claim to have liberated. The U.S.-led coalition, which has been conducting airstrikes in the offensive, said Tuesday that only a third of Fallujah can be described as "cleared," while other territory remains contested.

One of the militants' remaining strongholds is Fallujah's Jolan neighborhood, the northwestern corner of the city that was also the scene of some of the most persistent skirmishes between U.S. forces and insurgents in 2004 and 2007.

Its jumble of narrow streets and dense concentration of residential buildings is expected to make it harder to use airstrikes and Iraqi armor. Additionally, thousands of civilians are still believed to be trapped in the IS-held territory, according to the United Nations.

Capt. Muthhour Sabaar of the Anbar provincial police said his men fought alongside the Iraqi military's elite special forces as they pushed into central Fallujah last week from the south. Since then, his men have moved north and west.

"This fight will be our hardest yet," he said. The defenses in the remaining IS neighborhoods are expected to mirror what his forces encountered in the southern edge — the trenches, tunnels and fortified houses that are now shredded and collapsed from artillery fire and airstrikes.

Once his forces moved farther into the city from the south, there were no elaborate fortifications.

"Here, (IS fighters) just ran away. Honestly, there were no defenses at all," Sabaar said, gesturing to the Nazzal neighborhood just south of the main east-west highway that roughly divides the city.

The safest routes through the city snake in and out of main thoroughfares. As one convoy of armored vehicles moved along the main highway, special forces Corp. Ahmad Ahmad pointed to the liberated territory, including the Khalifa Mosque, one of the city's largest.

"It's cleared, but only from the outside. We don't know about the inside yet," Ahmad said. Iraqi forces advanced so quickly that teams specializing in defusing bombs were unable to keep up.

Along a road in central Fallujah, destruction is intermittent.

Sabaar, the provincial police captain, said the fighting on these streets was nothing like what he saw earlier this year in Ramadi, where hundreds of homemade bombs made progress painfully slow.

"Daesh believed this city was like a capital for them," Sabaar said, using the Arabic acronym for the Islamic State group. "They never thought Iraqi forces would be able to break the city's walls."

A row of opulent homes bearing graffiti reading, "Property of the Islamic State, not for sale or rent," was scorched black by smoke. Iraqi forces driving through the neighborhood say the buildings were used as an IS command centers, and the fighters set them on fire to destroy sensitive documents as they retreated last week.

Overall, the level of destruction in Fallujah appears to be less than in Ramadi, where Iraqi forces and the coalition were criticized because many parts of the city were mostly destroyed in the fighting. Iraqi and coalition officials countered that most of the destruction in Ramadi was caused by IS explosives.

The Islamic State's uneven defenses in Fallujah were probably the result of time and resources, said Aymenn Jawad al-Tamimi, a researcher with the Middle East Forum who studies the group through its documents.

The Fallujah militants "would have been distracted with fighting the Iraqi forces on the peripheries never too far away," al-Tamimi said.

Although Fallujah was under militant control longer than any other part of Iraq, it was only 40 miles (65 kilometers) west of Baghdad. IS was able to build defenses in cities and towns like Ramadi and Hit because of their relative isolation from territory under the Iraqi government's control, he said.

The one place in downtown Fallujah where IS put up a fight was at the central hospital.

Iraqi forces clashed with militants at the hospital for two days before making their way inside, Sabaar said. When they did, they found a half-dozen IS fighters buried in a shallow grave in the building's garden, in addition to the dead inside the hallways. Blood-soaked blankets filled one room where wounded militants presumably were treated. Hanging on the walls were IS-stamped notices forbidding bicycles and weapons inside the building.

The fight for the hospital took longer than expected because the coalition refused to bomb the building, which was empty of civilians and being used as a base by the militants, Sabaar said.

"I guess they thought it would look bad to blow up a hospital," he said.