Why not g4 after bh5?

often I will use h3 to push/kick the bishop and immediately follow with g4. I have noticed, however that this is not common among higher rated players. The knight is stilled pinned? 

It depends on the specific position. But if you go g4 you have probably removed the possibility of castling kingside safely or exposed the already castled king. 

Yes, makes for a tough time defending

Lol I would say it depends on the tomato. Sun ripened tomatoes are good with everything :-) 

Thanks for your analysis and the points you make. 

Capablanca made a comment about the maneuver a6 and b5 in the Ruy Lopez, kicking white's light square bishop.  He called it defensively weak but offensively strong.  I think the same thing applies to the kicking the bishop on the Kingside. 

You have to be careful about weakening the pawns in front on the Kingside if your king is castled there.  Good players know how to exploit such weaknesses so you may see it less in their games.

h3 is often a useful move anyway. It provides the castled king with an escape hatch and removes the risk of a back-rank mate. 

g4 has no such benefit. If you're castled kingside or plan to castle kingside, your king is now very easily exposed - g4 itself is a target for sacrifices, or it can be undermined with -h5 or -f5. 

Why h3 weakens the g3 square (which is still protected by the f-pawn, although now you need to be more cautious about that pawn getting pinned) g4 weakens h4 and f4. 

Yeah, like DS said it makes the bishop decide.

Notice that if it goes to h5 it's (almost every time) cut off from posting itself on 1/2 of the board (and it will probably be confined to the h and g files or trading itself on f3 for the rest of the game). Sometimes this is a really big decision. As a simple example squares like b6, c6, d5, and d7 become weak.

Here's a simple example where there's immediate disaster on black's queenside due to the bishop having moved away. The example includes the move g4, but the point is the queenside squares which the bishop had abandoned (the other two examples will not have g4).

Note that the bishop often goes to g4 in the first place to help fight for the center. However when that doesn't make sense, sometimes it's just a wasted move. In such a case if the bishop is challenged with h3 it should retreat back to the center or queenside.

And as for your g4 question, you'll notice in the example below that white does not play g4 because g4 would just slow down the attack.

And of course sometimes the queen's bishop belongs to white.

Thanks @stiggling. I like the move Nf1 and will try this. 

May you elaborate? Any piece? How can pawn on h3 prevent piece on h5? Thank you for clarification

I think main point is how to break the pin of Nf3 by Bg4?

There are many ways to break the pin.

1) play h3, Bxf3, Qxf3

2) play h3, g4

3) play Be2

4) play Nd2

5) remove the queen with a tempo ( say Qa4+)

6) remove the horse with a tempo ( say Ng4 with a threat of Bb5+ or legal mate type trap)

h3 is played as a way of knowing the opponent's intentions. Does he want to keep the pin or just take the piece or just testing us? If he wants to keep the pin, then other options are explored. g4 is playable but there are some risks involved in playing it:

1) g4 weakens the kingside and so, you have to now castle on the queenside or keep the king in centre.

2) Or if you are already castled on kingside, then your king is vulnerable after playing g4. Infact, that might be the perfect scenario to sac some pieces on g4.

Conclusion:

g4 is very playable if you haven't castled kingside yet. Infact, that would give you a headstart on pawn storming your opponent castle assuming he castled on kingside.

If you are already castled on kingside or plan to castle on kingside, then other options to break the pin might be more viable than g4.