Lyon paid eight million euros for the 31-year-old, who has 52 caps for the United States.
A 10 per cent profit-sharing interest on the capital gain of a potential future transfer is included in the transaction.
Turner's loan to the New England Revolution comes with a three-million-euro purchase option.
Turner moved to Forest from Arsenal in August 2023 and played 21 times in all competitions before heading to Crystal Palace in the 2024/25 season, in a year-long loan deal.
He helped Crystal Palace win their first major trophy after beating Manchester City 1-0 in the FA Cup final at Wembley in May.
Lyon also announced it had loaned Croatian international defender Duje Caleta-Car to Spanish club Real Sociedad for one year with a four-million-euro purchase option.
The 28-year-old Caleta-Car, who has 33 caps, arrived at Lyon from Southampton in 2023.