As mentioned, regardless of gauge the pellets will hit with the same power if going the same velocity, but pattern is the key. I killed several turkeys with a 20 before I could take a 12, and my brother did also. He shot one the farthest, about 32 steps, and it dropped immediately, but flopped around immediately and for a few minutes like it just wasn't quite nailed enough. We used both 3" 5's and 6's. I think 4's are restricting your pattern a bit much, as you are already restricted by going with a 20.

We used Rem 870 youth guns with standard full choke.