The 29-year-old goalkeeper has signed a four-year contract with the English club after just joining Arsenal last summer.
He served as backup to Aaron Ramsdale and made just seven appearances in only cup competitions.
His chances of getting more minutes became remote after Arsenal agreed to sign David Raya from Brentford earlier this week.
But Turner cemented his position as the first-choice goalkeeper for the USMNT and started all four games in the World Cup.
Arsenal will receive a fee of £10m for their backup goalkeeper who could face his former club this weekend.
“It’s a great challenge and a great step in my career,” Turner said. “When I heard that there was interest to bring me here, it just felt right for my family, it felt right when my agent spoke to me about it and it feels like the right move for me right now.
“The manager and the backroom staff, everything about the club has a little sparkle to it. It’s felt right from the start and I’m happy it’s got done. I’m looking forward to making a connection with my teammates, the fans and the city. I’m grateful to be here and can’t wait to get things started.”
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