Turner, a former No. 2 overall pick of the Philadelphia 76ers, spent the past three seasons with the Blazers.
Bazemore has been with Atlanta for the past five seasons, making him the longest-tenured current Hawk before the trade. At 29 years old, Bazemore had been the third-oldest member of the Hawks behind Vince Carter (42) and Miles Plumlee (30).
The teams announced the deal early Monday night.
Bazemore started 35 of the 67 games in which he played last season, averaging 11.6 points, 3.9 rebounds and 2.3 assists while shooting 32.