How do I brace a flexing 4" x 4" wooden gate post, suspending a heavy wooden gate from it?

have you got a brace on the gate that goes from the furthest point at top to bottom at start of gate ..at a diagonal ...you certainly should have ...and as you say you cannot fit a brace over the top of these gates all you can do is concrete a post stump in about 3 feet from gate and fit a diagonal brace when it has set.......ps answer no4 is right ...you would not normally fit such a small post to a gate ....here in uk we would use minimum 6x6 and probably 8x6