Nearly every night for a year, mortar and sniper fire from Islamic State group militants has pinned down outgunned Iraqi troops on the edge of Fallujah.

The city, the first to fall to the Sunni extremists a year ago this month, exemplifies the lack of progress in Iraq’s war against the Islamic State group, which holds a third of the country. U.S.-led airstrikes and Iranian aid have helped Iraqi troops, militiamen and Kurdish fighters take back bits around Islamic State-held territory, but recapturing it all remains far out of reach.

“We are constantly on alert and don’t sleep very much,” said Saad al-Sudani, an Iraqi soldier among the beleaguered troops outside of Fallujah. “We are waiting for any kind of support.” […]