I have with no issues, but your bow and arrow need to be reasonably well tuned. Shooting through some paper at the same distance as you shoot through your screens is a good test. Your broadhead will cut plenty of hole for your shaft and feathers to clear as long as your tune is reasonable. .
My only issue with shooting through screen is not arrow flight, but depth perception. It messes with me a lot. Not a big issue in bright light, but hard for me in low light early, late, and in the woods early season with the leaves all on. Not so bad for turkeys, but then as said by others, turkeys don’t care that much about the black hole so I don’t use the screens most of the time for them. For dear I tend to tree stand hunt or make sure the blind is brushed well and they keep the screens off my main hunting shot openings.