Turnesa was born in Elmsford, New York and lived most of his life there. His older brothers forbade him to turn pro and pooled their money to send him to college. He graduated from Holy Cross in 1938 and won his first U.S. Amateur later that year at Oakmont Country Club.
Turnesa won the British Amateur in 1947 at Carnoustie Golf Links, beating fellow American Dick Chapman 3&2. He won his second U.S. Amateur in 1948 and was runner-up in the 1949 British Amateur. He won numerous other amateur events, mostly in the New York area.
Turnesa played on three straight winning Walker Cup teams, 1947, 1949, and 1951. He was playing captain on the last team.
Turnesa served as president of both the Metropolitan Golf Association and New York State Golf Association. He co-founded the Westchester Caddie Scholarship Fund in 1956.
Turnesa died in Sleepy Hollow, New York.