Still, the Iraqi forces face hundreds of militants who appear determined to fight to the end in neighborhoods still teeming with hungry and increasingly desperate civilians.
The new operation will also require careful synchronization among the Iraqi Army, Interior Ministry troops and the Counterterrorism Service, which report to different authorities in Baghdad and sometimes seem to be fighting separate wars.
Tensions among Iraq’s disparate forces came to the fore in a closed-door meeting that Prime Minister Haider al-Abadi convened in mid-April in Hamam al-Alil, a town nestled on the Tigris River about 15 miles southeast of Mosul. Outfitted in the black uniform of the Counterterrorism Service, Mr. Abadi warned his commanders that prolonging the battle for the city would only play into the hands of the Islamic State and signaled that it was time to pick up the pace.
But that was followed by sharp debate among frustrated Iraqi commanders about which of their units have been making the greatest sacrifice and who should shoulder the burden of the next stage of the battle.
Lt. Gen. Raed Shaker Jawdat, the commander of the Federal Police, which have suffered many casualties as they advanced from the south along the Tigris, complained that his force was fighting hard but that the Iraqi Army had yet to push into a city filled with snipers and car bombs.
But Lt. Gen. Abdul-Amir Rasheed Yar Allah, the head of the operations center in Nineveh Province who is overseeing the Mosul battle, pointed the finger at the Federal Police, complaining that their once audacious assault was no longer gaining ground.
Underscoring the stakes, the meeting hall was decorated with the portraits of Iraqi troops killed in the campaign against the Islamic State.
“The point of the meeting was to ask if the Federal Police have done their job and how to get the army to participate more in the fighting inside the city,” said Lt. Gen. Abdul-Wahab al-Saadi, a senior commander with the Counterterrorism Service, who attended the meeting.
“Each of the leaders was trying to say his force had done the best, but let’s see the reality on the ground,” General Saadi added, asserting that his Counterterrorism Service had reclaimed the most territory from the Islamic State.
In the days of consultations with his fellow commanders that followed, General Amir appeared to have overcome many of the strains and built a consensus on a new plan. It gives the Ninth Iraqi Army Division a new role and has been quietly supported by Lt. Gen. Stephen J. Townsend, who commands the American-led task force that is fighting the Islamic State in Iraq and Syria.
The Ninth Division had positioned itself to the north of Mosul after fighting its way along a dusty ridge west of the city to cut the Islamic State’s supply lines and potential escape routes to the west.
The division’s southward push is being reinforced by troops from the Emergency Response Division, which answers to the Interior Ministry, and army brigades that had been pulled from security duties in eastern Mosul and near Tal Afar.
The theory is that attacking from the north will force the Islamic State to split its defensive operations and make it easier for the Counterterrorism Service to make headway from the south. The Federal Police are also holding their position in the southern part of western Mosul.
The operation has been timed to coincide with clear weather so that American Apache attack helicopters, armed drones and warplanes flown by the United States and allies can provide ready air support.
Once they clear much of the western part of the city, the Iraqi forces can turn their attention to the densely packed old section of Mosul, which still looms as the most difficult battlefield.
Col. John L. Dorrian, a spokesman for the American-led command in Baghdad, said on Wednesday that the Islamic State force in Mosul was “well short of a thousand fighters” but added that they were “going to be very difficult to get out.”
“When in that type of dense urban terrain, a small number of fighters can control that territory,” he added.
The Iraqi Counterterrorism Service was established by the United States soon after the 2003 invasion to conduct raids to capture or kill militants. It reports directly to the prime minister and has assumed a lead role in large-scale urban operations against the Islamic State because it is the best trained and most capable of Iraq’s fighting forces.
The Iraqi Army, which the American-led coalition has worked to rebuild after the loss of Mosul to the Islamic State in 2014, reports to the Defense Ministry. The Federal Police and the Emergency Response Division both report to the Interior Ministry.
“The organizations involved in the Mosul campaign have different reporting chains,” said David M. Witty, a retired Special Forces colonel with the United States Army and a former adviser to the Counterterrorism Service. “The various ministries that the units report to are competing fiefdoms, each struggling for power, influence, prestige and resources.”
The Iraqis struggled to synchronize their forces when they began their offensive to retake the eastern half of Mosul in October. But officials with the United States military insist that coordination among the Iraqi forces later improved with the help of American advisers.