Ammar Al Shamary and Gilgamesh Nabeel

Special for USA TODAY

FALLUJAH, Iraq — Iraqi military and militia forces rolled into this war-battered city Monday, capturing a police station and advancing the crucial campaign to drive Islamic State militants from one of their last major Iraqi strongholds.

"Our forces are still fighting in three directions in Fallujah," said Yahiya Rasul, spokesman of the Joint Operation Command. "The fight is intense."

Rasul said the push into the city was tentative as soldiers worked to clear mines and booby traps. "We are tightening the siege on the militants and advancing carefully," he said.

The government views capturing the Sunni-dominated city 35 miles from the capital of Baghdad as key to stopping a spate of deadly terror attacks over the past few months. Suicide bombers blew themselves up in three districts in Baghdad on Monday, killing at least 24 people and injuring dozens more.

Lieutenant General Abdelwahab al-Saadi, the commander of the offensive, said troops entered the city backed by air cover from the U.S.-led military coalition. The U.S. provides extensive support for Iraq’s military, including providing arms, surveillance, intelligence and advisers.

Iraqi Prime Minister Haider al-Abadi said in a televised speech to parliament that the "current phase" of the battle could be completed in two days.

Iraqi forces initially surrounded Fallujah more than a week ago, cutting off Islamic State from supplies and reinforcements after taking much of the region around the city's center.

Iraq hasn't asked for extra help in Fallujah

Iraqi military starts to retake Fallujah

Fallujah was the first city to fall to the militants when they swept across northern and western Iraq in 2014, bent on building a fundamentalist Sunni caliphate from a wide strip of Iraq and Syria. But Islamic State fortunes waned in recent months, and Fallujah and Mosul are the only major Iraqi cities still under the militants' control.

Conditions in the city steadily deteriorated over the past few months, and the United Nations issued a report in April saying residents face acute shortages of food and medicine.

Iraq’s government has been under political pressure from Sunni leaders to liberate Fallujah, where tens of thousands of people, many of them Sunnis, still live despite years of fighting. Recently, there have been reports of civilians starving in the city, which has increasingly been cut off from the rest of the country.

The Islamic State forced civilians caught in the city to serve as "human shields" in areas where the militants expected airstrikes, residents recently USA TODAY.

Abu Mohammed, 24, an optician in Erbil, says his aunt and her family are still trapped in the city, with little to eat and no electricity or water.

“Living conditions are so miserable," Mohammed said. “My aunt said that they are eating dates and yogurt for the past month. People go out shopping when it is calm and there is no bombardment.”

“People cannot go to markets, and when they do, the prices are increasing, with the price for a flour bag doubling 10 times," he added. "Vegetables are available now, provided from what is planted inside private home gardens – still only those who live by the river can plant vegetables as the water supply has been cut.”

Nabeel reported from Istanbul. Contributing: John Bacon in McLean, Va.