After making rapid gains in a new offensive, Iraqi forces are close to choking off the last bastion of several hundred Islamic State fighters dug into the twisting alleyways and narrow streets of Mosul’s Old City.

The final push will be led by Iraqi’s U.S.-trained Counter Terrorism Service units, the same forces that first entered the eastern reaches of the city in October when the battle to retake the city began, officers said. The plan involves clearing the final few neighborhoods north of the Old City and surrounding the square-mile-wide neighborhood.

With tens of thousands of civilians packed into an area defended by Islamic State fighters, snipers and car-bombers, the battle could quickly turn into a humanitarian crisis.

Some Iraqi military commanders say they are trying to finish the battle in the next two weeks, before the holy month of Ramadan begins and many soldiers will begin fasting during the day. Yet with the Islamic State still doggedly fighting for each block, it is unclear whether the campaign will end any time soon.

Lt. Gen. Sami al-Aridhi, the commander of two counterterrorism task forces, estimates that there are between 250 to 350 Islamic State fighters, many of them French and Russian, in the Old City, along with more than 200,000 civilians.

Some of the counterterrorism troops, however, estimate the Islamic State presence at more than 1,000 fighters, given the stiff resistance they are still facing on the ground.

[Away from Iraq’s front lines, the Islamic State is creeping back in]

The Old City holds symbolic significance: It was here, at the Great Mosque, that the Islamic State’s leader, Abu Bakr al-Baghdadi, declared a “caliphate” in 2014 across parts of Iraq and Syria.

The neighborhood also presents a heightened challenge for the units. Known as the CTS, the force was originally formed to work as an elite raid force after the U.S.-led invasion in 2003 but gradually turned into the Iraqi military’s premier offensive infantry unit after the rise of the Islamic State. Aridhi and other officers said they are prepared to attack the Old City but will have to adjust some of their tactics as they clear the tiny streets and alleys.

In the past, the advancing counterterrorism troops have fought primarily from their vehicles, block by block, methodically using their Humvees and armored bulldozers to seal off a street before taking the surrounding buildings on foot in an almost grid-like fashion. As a block is cleared, civilians often stream out of their homes, slowing the advance while troops usher them to safety.

Lise Grande, the U.N. humanitarian coordinator for Iraq, said in a statement Thursday that the United Nations expects more than 200,000 people to flee as Iraqi troops close in on the Old City, a figure she called “alarming.”

[How a woman in England tracks civilian deaths in Syria, one bomb at a time]

In the Old City, the modified D-8 bulldozers that have been essential in stopping car-bombs and clearing obstacles set up by the Islamic State will be all but useless, likely forced to remain behind on main thoroughfares while the commandos move on foot. The Islamic State has increasingly dispatched motorcycles rigged with explosives and is expected to use them extensively in the Old City.

Brig. Gen. Ali Jamal, the head logistics officer for the counterterror forces, has requested smaller bulldozers from the U.S.-led coalition but has received only three, all lacking armor, he said. Since the beginning of the campaign, his forces have lost eight of the tracked vehicles, their twisted hulks sometimes dragged back to headquarters and abandoned in other parts of the city. On Tuesday, three bulldozer drivers were wounded by car bombs and snipers, forcing the troops to pause their advance until fresh ones could be sent from other units.

One of the wounded drivers, Sgt. Mohammed Ali, rammed a suicide vehicle while helping clear the Ar Rafa’l neighborhood. Just before the car exploded, Ali pushed it toward a nearby wall, a moment he captured in a cellphone video that went viral. He received only minor injuries from the shrapnel and spent the next day behind the lines showing the video to reporters.

In addition to the loss of the bulldozers, the counterterror forces will likely have to rely less on coalition airstrikes to clear the Old City. U.S. and other coalition forces have been accused of killing thousands of civilians during their air campaign over Iraq and Syria, and claims of errant strikes have only increased since Iraqi forces entered Mosul.

The air support has been essential in helping the counterterror forces push through the city without suffering heavy casualties, a necessity as the unit is expected to continue clearing parts of western Iraq after the fall of Mosul, according to coalition advisers who work with the unit.

With the Islamic State’s propensity to use human shields and the sheer number of civilians left in the Old City, any type of airstrike will have to be heavily vetted to ensure to only Islamic State fighters are targeted. In March, a U.S. airstrike, launched at the request of the counterterror forces, killed upward of 100 people in the al-Jadida neighborhood of Mosul. The coalition is still investigating the incident.

Col. Arkan Fadhil, a battalion commander who works with U.S. and Australian special operations troops to help call in airstrikes, said the counterterror forces will probably lean more heavily on preplanned targets before they start advancing into the Old City. It is unclear how that will work in practice, especially if the Islamic State attempts to take advantage of the area’s close quarters and tries to isolate and overrun small groups of advancing Iraqi forces.

“We’re ready to go into the Old City,” Arkan said. “We’re just waiting on a tasking order.”

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