King vs Pawns .... 5 examples

The key is even though black is 'closer' white gets to the key squares of b6,c6, or d6 before black gets to f5, g5 or h5.  Black cannot defend passively with Ke7 either as white gains opposition on the 7th rank.

White can answer Kh1 g4 with Kg2 gxf3+, Kxg3 Kd2, Ke4