The team's quarterback Brandon Weeden broke into the NFL last season following a short career in professional baseball, having been chosen as the 22nd overall pick in the 2012 draft.
The 29-year-old was criticised for his performances last season, throwing more interceptions than he did touchdowns, and Turner admits that the offense has been too inconsistent for his liking.
Asked whether Weeden would be usurped as quarterback, Turner said: "I don't envision that happening because of the way he's playing, but we've got to go play.
"That's a question more for [coach Rob Chudzinski] than for me, but I think everyone in that locker room understands that we're committed to playing the guys that give us the best chance to win. Until a guy has started and played at a high level and won, there's always going to be competition.
"We've had a lot of mixed results and more inconsistencies than I'd like, but that doesn't surprise me. However, we do need to have that sense of urgency that this thing's going to be on us real fast."
Cleveland finished bottom of the AFC North last year, finishing the season with a 5-11 record.