Cats are hunters. Most cats would have killed the prey (and many will lay it out for you).
You can help the prey instict by toys that involve catching (a wand with a toy on a string and a lazor works well). But the cat's instincts is to catch real prey. Its not your fault or the cat's fault. Its just what most cats do.
Can you re-enforce the screen? Im less worried about the birds then I am about the cat going outside by the way of the window or falling many stories if up high. Cats also have a natural instinct to roam but it is much, MUCH healthier and safer inside. Outside cats, on average, has a 3 yr life span.
Is there any way to fix the screen so it wont open at all? Is it a pop out screen? What about nailing some laticework on the window or nailing something like that inside so if the screen pops out you have laticework or chicken wire or something nailed inside the window sill? Is that doable?