The Germany international was deemed surplus to requirements at the Emirates Stadium this summer, having spent the second half of last season on loan at Inter, where he scored just one goal in 17 Serie A appearances.
Now in Turkey, Podolski is hoping to reignite his career and help Galatasaray claim major silverware.
"Galatasaray also made proposals in the past year," Podolski said. "But this year there was more intense interest. My parents wanted me to come here.
"Galatasaray were champions last year. We will play to be champions in the coming years, we're looking to get to the next level.
"I've followed Galatasaray for a long time. I know there's a great atmosphere in the stadium.
"Galatasaray fans have given me so much attention and are excited about me. This is a big factor in my decision-making."