For turkeys, I'd recommend no screen and wearing black hat, shirt, gloves inside the blind. In my limited blind experience, deer don't like open windows, but turkeys aren't usually bothered by 'em.
I've posted here and elsewhere before about whether a non-centershot traditional bow, launching an arrow that might be going through some archer's paradox when it hits the screen, and then fly off course... And every time the veterans tell me it doesn't happen. Yet I've always felt what happened to you, could happen frequently. I've just not taken the time to practice much to find out.