Turner is headed to his fifth team in 10 seasons since he was drafted with the second overall pick in the 2010 draft by the Philadelphia 76ers.
"We are happy to add Evan to our team, a veteran who we believe can help our club," Hawks general manager Travis Schlenk said in a statement. "The versatility he has shown throughout his career will be valuable for us this season."
Portland has been shopping for insurance in the event free agent Rodney Hood cannot be re-signed.
Turner, 30, averaged 6.8 points, 4.5 rebounds and 3.9 assists per game in 73 games with Portland last season. His base salary for the upcoming season is $18.6 million, the final year of a four-year deal Turner signed with Portland.
Turner has career averages of 9.9 points, 4.7 rebounds and 3.5 assists in 686 career games.
Bazemore, also heading into the final year of his deal, is due to make $19.3 million in 2019-20.
Bazemore, 29, entered the NBA as an undrafted free agent with the Golden State Warriors. He also played for the Los Angeles Lakers before spending the past five seasons with Atlanta. He averaged 11.6 points, 3.9 rebounds and 2.3 assists per game last season (67 games played).
"Kent is a two-way wing that adds depth to our perimeter, is an excellent fit with our returning players and will transition seamlessly to our style of play," Trail Blazers president of basketball operations Neil Olshey said in a statement. "We want to thank Evan for his many contributions to the team. He was a critical player in our run to the Western Conference Finals and he will be missed on and off the court."
"We are appreciative of Kent's contributions to the Hawks organization and to the greater Atlanta community over the last five years," Schlenk said. "We wish him and his family all the best in Portland."
Bazemore's career scoring average is 8.7 in 483 NBA games.
--Field Level Media