Turina started his professional career with his youth club Dinamo Zagreb in 1998. Before settling as a first-team member in 2003, he completed loan spells at Croatia Sesvete, Kamen Ingrad and Osijek. He was Dinamo Zagreb's first-choice goalkeeper during the 2005–06 season, when the club won their first league title in two years. He continued to play regularly in the early stages of the following season. However, after Dinamo Zagreb were eliminated from the UEFA Cup with a 5–2 aggregate defeat to AJ Auxerre in the first round, Turina was replaced in the starting line-up by Filip Lončarić. At the time, he was criticized for letting a long-range shot by Ireneusz Jeleń slip through his hands when the French club took a 1–0 lead in the first leg in Zagreb. Following the signing of German goalkeeper Georg Koch in the summer of 2007, Turina decided to join Greek club Skoda Xanthi, spending one season with them before leaving for Polish club Lech Poznań.