Iraq's Shia militias have withdrawn from the forefront of an offensive to recapture Tikrit from Islamic State jihadists, allowing Baghdad government troops to take the lead, a top US general said Thursday.

The Shia militias "have pulled back from that area" and Iraqi special forces and police are "clearing" Tikrit, General Lloyd Austin, head of US Central Command, told lawmakers.

Austin confirmed previous comments by officials that a US condition for air strikes against the IS group in Tikrit was that the Iraqi government be "in charge" of all forces in the assault on the town.

The four-star general said the offensive to take back Tikrit launched on March 2 had been flawed because some of the forces in the operation were "not supervised" by the Iraqi government.

Washington says Iranian military officers have provided advice and artillery to Shiite militias involved in the operation. And the White House has been dismayed by Tehran's prominent role in the assault, which was launched more than three weeks ago amid triumphant statements from Baghdad.

But as the offensive bogged down in recent days, President Barack Obama's administration seized an opportunity to exert US influence when Baghdad issued a request for American air power.

Austin said about 4,000 Iraqi army and federal police forces were leading "clearing operations" in Tikrit, the hometown of executed dictator Saddam Hussein.