The 25-year-old will move to Old Trafford in a deal worth £4.2 million ($5.3m). He has signed a contract until 2027 with the option of another year.
"It is a huge honour to join Manchester United and become the first Turkish player to represent this incredible club," Bayindir said.
"I have a passion for success, and I will give everything to help this special group of players to achieve our ambitions.
"I cannot wait to start working with such an experienced goalkeeping unit. We will support each other and drive high standards every day so that each of us is ready to perform whenever called upon."
Tom Heaton will also be part of Erik ten Hag's squad this season but the 37-year-old is injured.