BAGHDAD—Ask anyone in Baghdad’s Sunni Muslim neighborhood of Aadhamiye about life under Iraq’s Shiite majority, and you’re likely to get a puzzled reply: “What Shiite majority?”

“Sunni Arabs aren’t few. We are at least half the population of Iraq,” said Fares Ali Karim, a 55-year-old who runs a travel agency near the Aadhamiye mosque, where Saddam Hussein made his last public appearance as president in 2003.

That...