B.P.
At that age, I don't think the lessons are really swimming lessons as much as not drowning lessons. It sounds to me like he's doing great.
Like the others have said, he has a comfort zone with his instructor. You might contact the instructor for some verbal help and go to a friend's house or public pool to work on that.
I know when I was young I did swimming lessons for 2 years before I had it all down.