AMIRIYAT FALLUJA, Iraq — On one side of a rickety bridge that spans a narrow stretch of the Euphrates River were panicked families on the run from Islamic State forces, hoping to escape Anbar Province and reach safety in Baghdad. On the other side were Iraqi Army officers and Shiite militiamen, under orders to keep the bridge closed because of fears that militants could slip in among the displaced civilians.

“It’s like the other side is Europe and this is Asia,” said Ehab Talib, 27, who was waiting to meet relatives fleeing the fighting in Anbar, the Sunni-dominated region whose capital, Ramadi, recently fell to the Islamic State.

With new waves of civilians fleeing violence in Anbar there are now more internally displaced Iraqis, nearly three million, than there were at the height of the bloody sectarian fighting that followed the American invasion, when millions of Iraqis were able to flee to Syria. That door is closed because of that country’s own civil war. And now doors in Iraq are closing, too, worsening sectarian tensions as the Shiite authorities restrict where fleeing Sunnis can seek safety.

“We are all Iraqis,” said Marwan Abdul, a doctor’s assistant, standing outside his mobile clinic here. “This wouldn’t happen in any other country.”