BAGHDAD, Iraq — As the bullets whizzed past First Lieutenant Ben Hartig, he scrambled for cover and glanced up at a giant Sadr City billboard of the anti-American cleric Muqtada al-Sadr.

“We’ve killed so many of his Mahdi Army militia guys, but they just keep shooting and shooting,” Lt. Hartig, 23, of Concord, Calif., yelled as he struggled to be heard over the cacophony of Bradley Fighting Vehicles firing back with 25-mm cannons. “Sadr is a really tough enemy.”

But not officially. In Washington and in the capital’s heavily fortified Green Zone, American military and diplomatic officials rarely mention Mr. Sadr and his army when describing their enemy. They say American troops are clashing with “criminal elements” or “special groups,” a phrase used by the military to refer to Iranian-backed fighters.

The facts on the ground in Sadr City, the vast Shiite district that is Mr. Sadr’s Baghdad stronghold, tell a different story. According to American infantry soldiers, Mahdi Army fighters, and the American brigade commander here, the conflict in Sadr City is essentially a fight between terrorists loyal to Mr. Sadr and the American and Iraqi militaries.

The delicacy of the American terminology underscores both the fragility of the security gains in Iraq and the American government’s efforts to tie Iran to the ongoing violence. For U.S. soldiers, the battle is more straightforward.

“Of course we’re fighting JAM,” the commander of the brigade leading the offensive in Sadr City, Colonel John Hort, said referring to Jaish al-Mahdi, which is Arabic for Mahdi Army. “There are hundreds of them throughout Sadr City, and we’ll keep up the fight against them until they stop attacking us.”

According to American officials, every bona fide member of the Mahdi Army is obeying Mr. Sadr’s cease-fire, and any member fighting American or Iraqi troops is by definition violating his leader’s order and therefore a rogue element.

One of the American military’s top targets in Sadr City is the senior Mahdi Army commander in the enclave, Tahseen al-Freiji. Colonel Hort said the goal was to remove Mr. Freiji and other top targets as threats, either by detaining or killing them.

American officials said that Mr. Freiji commands a full brigade in Sadr City, directing 6,000 to 8,000 men. They said he has given orders to launch rockets and mortar shells; fire rocket-propelled grenades and AK-47 assault rifles; and set roadside bombs.

[Elsewhere, the Associated Press reported that the Senate yesterday kicked off debate on legislation to add a grab bag of domestic programs to President Bush’s war request, including work permits for immigrant farm labor and heating subsidies for the poor. Majority Leader Harry Reid, a Democrat of Nevada, brought up the add-ons in an unusual move designed to win their adoption over opposition from the White House and Republicans. The new GI Bill and Democratic priorities like extending unemployment benefits are among the big-ticket add-ons, both of which have drawn veto threats.]