The two sources I'd depend upon to answer this question are actual games played by GM's (best), and engine analysis, carried to a depth of at least 24 ply (12 moves by each side).
Here's what I get:
Actual Games
Move Games Win % Latest Players
4 d4 2947 57.3 2016 Carlsen, Kasparov, Karjakin, Anand
4 Bb5 513 57.8 2016 Grischuk, Karjakin, Morozevich, Adams
4 Bc4 229 47.2 2016 Vallejo Pons, Movsesian, Lu SHanglei, Zhigalko
4 g3 153 48.4 2010 Mamedyarov, Romanishin, Galego, Biyiasas
4 h4 38 43.4 2012 Hamdouchi, Pilavov, Minasian, Pridorozhni
So, the move is pretty unpopular at the highest levels (about 1% of games), and since Black outscores White by 57%-43% in this line, it's no wonder.
Engine Analysis
Engine parameters: Stockfish 7, QuadCore AMD CPU using 4 Cores, Depth=25/39
Line Move Eval
a. 4 Bb5 0.17
b. 4 d4 0.13
c. 4 h3 0.13
d. 4 Rb1 0.11
e. 4 a4 0.10
f. 4 Bc4 0.09
g. 4 a3 0.03
h. 4 Be2 0.01
i. 4 d3 0.00
That's 9 lines that are better than 4 h4, and the 9th has an evaluation of dead equal, implying that the move 4 h4 must have a lower evaluation. Note that 4 h4 starts at among the top four moves until the engine reaches a depth of about 15-17 ply, and then falls off the map. So the blowback seems to occur somewhere between move 11 and move 13.