The overnight raids were launched in the sprawling Sadr City in eastern Baghdad, a stronghold of the Mehdi Army militia loyal to firebrand Shi'ite cleric Moqtada al-Sadr.
"Iraqi forces quickly secured the insurgent's residence in Sadr City and detained 10 suspected death squad members," the U.S. military said in a statement late on Wednesday.
It said the Iraqi special forces came under fire and called in U.S. air support, killing 10 "enemy fighters."
Iraqi forces then moved through Sadr City to a mosque where the kidnapping suspects were believed to be located and detained three suspects, it said.
The U.S. military first reported that the target of a raid in Sadr City was a death squad leader. A military spokesman said no air strikes had been conducted in the hunt for the U.S. soldier of Iraqi descent, who went missing on Monday after leaving the safety of the Green Zone.
A search operation is under way for the soldier, who is assigned as a linguist, and U.S forces have set up checkpoints and conducted house-to-house searches in central Karrada area.
The Iraqi Interior Ministry said four people were killed and about 20 wounded.
Asked about the raid at a news conference earlier on Wednesday, Prime Minister Nuri al-Maliki appeared to distance himself from it, saying there was a lack of coordination. "We will be seeking a full explanation from the multi-national forces," he said.
Sunni leaders and U.S. officials say the Mehdi Army is behind sectarian killings and kidnappings that have pushed Iraq to the brink of civil war. U.S. ambassador to Iraq Zalmay Khalilzad said on Tuesday it had to be brought under control.
A resident told Reuters two members of the Mehdi Army were among the dead. Angry relatives and the wounded blamed both U.S. forces and Maliki's government.
"Where is Maliki? Where is his freedom?" said one man lying on a stretcher in the hospital.
The search for the missing soldier revived memories of the kidnapping in June of two U.S. soldiers near Yusufiya, an al Qaeda stronghold south of Baghdad, in an ambush. Their bodies were later found badly mutilated.
The soldier was last seen on Monday afternoon in the Green Zone, in uniform. A military spokesman declined to say whether his superiors were aware of his plan to visit a relative.
"He was kidnapped outside the International Zone. Everything else is under investigation," Lieutenant-Colonel Steve Stoder said.
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