WASHINGTON—Federal prosecutors have dropped charges against 11 of 15 members of Turkish President Recep Tayyip Erdogan’s security team who were accused in connection with the beating of protesters during their visit to Washington last year, the latest twist in a case that caused a diplomatic rift between the U.S. and Turkey.

The decision by the U.S. to prosecute the 15 men added to political strains as the Trump administration was trying to reset relations with Turkey, a key U.S. ally in the fight against Islamic State. The...