BAGHDAD—Behind the rise of a paramilitary force in Iraq credited with saving the country from Islamic State is an Iran-trained jihadist the U.S. wants far from the battlefield.

A decade ago, the Iraqi known to U.S. officials as Jamal Jaafar Ibrahimi was in a cat-and-mouse game with U.S. forces and on the run from a Kuwaiti death sentence for allegedly orchestrating bombings at the American and French embassies there in the 1980s. The U.S. Treasury lists him as a terrorist.

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