Tips on keeping window screen in window sill
Without knowing exactly how the screen fits into the window, I'm not sure that this will work, but here's what I did (I had exactly the same problem - the cats learned that they could get out by pushing against the screens.)

In my case, the screens had an aluminum frame - I took a few small finishing nails, and gently hammered then through the frame, into the channel in the window frame.  I made sure that there was a bit of play in the nail so that they could be pulled out in an emergency (some of the windows were in bedrooms).