The goalkeeper made two dreadful errors in Liverpool's UEFA Champions League final loss to Real Madrid in May.
The Reds subsequently splashed out €75 million ($117.7 million) on Brazil international Alisson from Roma, a world-record transfer fee for a goalkeeper at the time.
Karius has now been allowed to leave and rebuild his career in the Turkish Super Lig side after agreeing a two-year loan to Besiktas.
Karius was named on the bench for Liverpool's Premier League opener, a 4-0 defeat of West Ham United, but was omitted from the squad for the subsequent trip to Crystal Palace and Sunday's (AEST) win against Brighton and Hove Albion.
The German made 49 appearances for Liverpool after arriving from Mainz in 2016.
"Everybody at LFC wishes Loris the best of luck during his loan spell," a Liverpool statement read.
The move came as little surprise after Besiktas teased the goalkeeper's arrival by mistake on its Twitter account.