BAGHDAD — Iraqi security forces and allied Shiite militias seized large parts of Tikrit on Tuesday, amid reports that most of the Islamic State militants battling to hold the city had begun retreating, security officials said.

The progress came after a week of heavy fighting to retake Tikrit, a city in the so-called Sunni triangle that holds strategic and emotional importance in the effort to roll back the Islamic State’s lightning advance toward Baghdad in June.

The offensive is the largest pro-government military operation yet, involving a combined force of more than 30,000. And if it succeeds, it would be a significant step in the march north to Mosul, Iraq’s second-largest city and an early conquest for the Islamic State, also known as ISIS or ISIL.