I live right down the road from a state game area that has released birds.
They are very meaty.
Two suggestions come to mind after you pattern your shotgun.
One is consider the range that you are seeing the birds. that they could be further out than you think. The other thing would be you might be jumping too far ahead of the bird on the flush and barely catching them with the fringe.
Both are easy to do, especially if you don’t shoot a lot of them.
Shooting pheasants in the front half takes a bit of practice, and a little bit of slowing down.
You really can, insert the shotgun behind the flushing bird, and then stroke through the front of the animal and fire.
To great effect.