TIKRIT, Iraq » One week after the start of intensive U.S. airstrikes against Islamic State hide-outs, most of the key parts of central Tikrit had finally fallen to the government’s forces — although significant pockets remained contested on Thursday.

But to hear some of the Iraqi forces here tell it, the Americans deserve little or no credit. And many of the Shiite militiamen involved in the fight say the international coalition’s air campaign actually impeded their victory — even though beforehand they had spent weeks in a stalemate with militants holed up in Tikrit. Some even accuse the United States of fighting on the side of the Islamic State.

Still, most of the militiamen now pouring into this city in the Sunni heartland along the Tigris River were not even in the real battle over the past week, and the only shots they fired were into the air on Thursday — which they did with abandon.

One militiaman interviewed on Thursday, Ali Jawad, a 36-year-old Badr Organization veteran from Hillah, was a little more battle-hardened than most. He was here throughout the fight and lost two friends on the front line near a former palace of Saddam Hussein, the last major building in Tikrit’s sprawling palace complex to fall to Iraqi forces. That happened, Jawad said, late Wednesday — even after Prime Minister Haider al-Abadi took a carefully choreographed victory stroll through downtown Tikrit.

Jawad, who had the faraway look of an infantryman after a long battle, was scornful of wasting precious ammunition firing into the air. "It was a hard fight," he said, "and we did not have anything to waste."

Still, he saw nothing to thank the Americans for, he said, even though he had watched their airstrikes with satisfaction. When it came to the final victory on the palace front, Islamic State fighters for the most part just fled, he said.

"This is the victory of Hadi al-Ameri and God," he said. "The Americans had nothing to do with it." Al-Ameri is the leader of the popular mobilization forces, as the militias are collectively known, and is a pro-Iranian official who was outspoken in his opposition to the U.S. involvement in the fight.

Jawad’s comrade, Mohammad Takrif, 24, a car mechanic from Hillah, had a similar take. "Thank God and Hadi al-Ameri, and also the Iranian advisers who helped us," he said. And the Americans? "All they did was bomb the wrong side and kill federal policemen the other day."

He was referring to a reported airstrike on Friday. But senior Iraqi generals now say that was a so-called friendly fire accident that they attributed to planes from the tiny Iraqi air force, not the U.S.-led coalition.

Another militiaman, Adel Mehsan, 40, a driver from Hillah who has also been in the thick of fighting around Tikrit for months, went even further.

"The Americans supported Daesh, not us," he said, referring to the Islamic State by its Arabic nickname. "I saw them dropping supplies to Daesh with my own eyes."

To some extent the fighters may simply be picking up the political line of their leaders. Many of the Shiite militias had fought the Americans during their pre-2011 presence here, and have strong links to Iran as well as to Iranian advisers.

Muen al-Khadimy, one of the popular mobilization’s top leaders and a senior official in the Badr Organization, a prominent Shiite militia, toured Tikrit on Thursday with an odd entourage of fighters and supply vehicles, including a pickup truck loaded entirely with tubs of date pastries and dozens of artists’ easels.

"Yes, the international coalition helped, but not really in a good way," he said. "Without them, we would have liberated Tikrit by Tuesday. They caused a big confusion with our plan."

The mayor of Tikrit, a medical doctor named Omar Tariq, said Thursday that 95 percent of the city was liberated "and we will finish everything else today." Only parts of the Qadisiyah neighborhood remained to be subdued, and some Islamic State fighters fled to a suburb, Alam, while elsewhere soldiers were still doing house-to-house searches, he said.

Tariq, whose hospital lies in ruins after serving as an Islamic State fighting position, ticked off the reasons he thought Tikrit was finally falling: "Good preparation by Iraqi forces. Cooperation between Iraqi forces and civilians in the area. The popular mobilization. Good intelligence."

Then, almost as an afterthought he added: "Also the airstrikes. But that was thanks to Iraqi cooperation."

Just how much of Tikrit really has been liberated remained unclear even Thursday, as guides from the Badr Organization broke off a planned tour of the city, other than visits to the clearly liberated palaces, and to the commercial center of the town, which lay in such ruins that there were scant pickings for looters — although not a few of the militiamen could be seen looting nonetheless.

Journalists were not free to roam the city, and even a promised visit to a mass grave of Iraqi air force cadets was canceled without explanation. Human rights groups and Iraqi authorities believe that 1,700 unarmed cadets from nearby Camp Speicher air academy were massacred by the extremists after they were sent home by their academy in the midst of the Islamic State advance last June.

There were no Islamic State prisoners taken at all in the recent fight, said Khadimy, the senior Badr official.

"To be honest, everywhere we captured them we killed them because they were the enemy," he said. Then, perhaps realizing how that sounded, he explained that any Islamic State fighters who were about to be captured were assumed to be suicide bombers, so they were killed as a precaution.

A battalion commander of the popular mobilization forces who was involved in the fight here for the past several months said that this week his men had captured three Afghan men, an Afghan woman and an Algerian man, all Islamic State fighters, in the battle for the palaces.

"After we were done with them, we killed them," he said, speaking on the condition of anonymity because he did not want to be identified admitting to a war crime.

Even at higher levels of the Iraqi establishment, whatever praise was emitted for the U.S. role was grudging at best.

Al-Abadi, for instance, credited "the joint efforts of the army and police forces alongside the popular mobilization fighters and the tribal fighters and the people of Tikrit with air coverage of the Iraqi air force and the international coalition."

The Iraqi air force has all of a dozen attack jets, but less than half are known to be in service, and none are equipped for precision bombing.

Al-Ameri himself, visiting here Thursday, was gracious but not grateful.

"We respect the decision of our government and the prime minister," he said when asked about the U.S. role. "The Iraqi government will thank them, since they’re the ones who called them."