The US international has joined on a four-year deal after spending 12 months with the Gunners, where he solely featured in cup competitions.
The 29-year-old made his international debut in February 2021 and has played 32 times for his country. Turner joins compatriot Ethan Horvath as part of Forest’s goalkeeping cohort.
“It feels great to be here,” Turner told Forest’s official website. “It’s something I’m really looking forward to, it’s a great challenge and a great step in my career.
“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 team-mates, the fans and the city. I’m grateful to be here and can’t wait to get things started.”