Lyon paid eight million euros for the 31-year-old Turner, who has represented the United States in 52 international matches.
The transfer agreement includes a clause that gives Lyon a 10% share of any profit made from a potential future sale of Turner.
The loan deal with New England Revolution includes a three-million-euro option to buy Turner's rights.
Turner previously played for Arsenal before moving to Nottingham Forest in August 2023. He made 21 appearances for Forest in all competitions before joining Crystal Palace on a year-long loan deal in the 2024/25 season.
During his time at Crystal Palace, Turner helped the team win their first major trophy by defeating Manchester City 1-0 in the FA Cup final at Wembley Stadium in May.
Lyon has also loaned Croatian defender Duje Caleta-Car to Spanish club Real Sociedad for one year. The 28-year-old Caleta-Car, who has 33 international caps, joined Lyon from Southampton in 2023.
