TIKRIT, Iraq — Iraqi officials insisted for weeks that Islamic State fighters had been all but exterminated in Tikrit, confined to a few pockets in the city center. Yet on Sunday, military officials in the city were reluctant to allow journalists to head back to Baghdad by road — even though the highway skirts Tikrit well to the west.

The supposed safer alternative was a general’s Iraqi Air Force Cessna waiting at the Tikrit Air Base nine miles northwest of downtown. But before takeoff, two mortar shells slammed into a grassy patch between the airfield’s two runways, within 100 yards of the small plane. Iraqi military escorts surmised that the person shooting had to have been within visual range — and probably to the west, although downtown was southeast.

“Daesh are everywhere,” one senior officer said, using the Arab nickname for the Islamic State, also known as ISIS or ISIL.

During a two-day visit to Tikrit, a strategic city in Iraq’s central Sunni heartland, it was clear that after four weeks of the government offensive the Islamic State’s fighters are more numerous and still hold much more territory here than officials had previously allowed, even with heavy American airstrikes added in.