The Nigeria midfielder, 23, has signed a four-year contract after leaving Italy for personal reasons.
"I needed to make this move to clear my head and be happy playing football again," he told BBC Sport.
Onazi has dismissed worries about security in Turkey, despite a failed coup against President Recep Tayyip Erdogan in July.
"Personally, I think security is the same everywhere," Onazi added.
"Nowhere is perfectly safe in life, so we just keep praying that God will guide and protect us wherever we go."
Onazi spent five years with Lazio, who he joined from Nigerian club My People, and scored seven goals in 110 appearances.
Injury and lack of form hampered him last season, and he almost left in January, when a proposed loan move to Bundesliga side Werder Bremen collapsed.
He played 20 games for Lazio during the campaign, although he became a more regular starter when Simone Inzaghi took over as coach in April.
"I left Lazio for personal reasons," said Onazi, who had two years left to run in his contract at the Stadio Olimpico, and had been offered a deal to extend it.
"It might be difficult for some people to understand my decision but this is a step forward in my career.
"The Lazio coach spoke to the management about keeping me. I was even offered a contract extension to stay but Trabzonspor provides a new lease of life.
"I will always be a Lazio fan because the club gave me the platform to be the player I am today."
Former Lazio and Nigeria striker Ayodele Makinwa, the midfielder's agent, said that Onazi would benefit from the move.
"He desperately wanted a change and new motivation," said Makinwa.
"Trabzonspor is one of the biggest clubs in Turkey and it is a stepping stone towards achieving his goals as a professional footballer.
"Joining a club fighting for the title will challenge and push him to the next level.
"He will definitely be one of the leaders in the team and this should serve as a motivation to achieve what he wants."
Onazi, who has 38 international caps, helped Nigeria win the 2013 Africa Cup of Nations, and played in all four of his country's games at the 2014 World Cup in Brazil.
Ogenyi Onazi says he has moved to Turkey "to clear my head" after joining Trabzonspor from Lazio.