If MOSFET is a voltage-controlled device, then why do we need to supply it with high current when using in an H-bridge?
Exactly what you suspect. The effect of the gate capacitance is to slow down the switching. (The Miller effect multiplies the 'effective' gate capacitance.)

If we want the H-bridge to switch only occasionally (let's say at 1 Hz) a low gate current is (in most cases) fine, because the thermal effects of the switching are spread over 1 second, which is a relatively long period. 

But if PWM is used, for instance at 300 Hz, there is only ~ 3ms to spread the heat dissipated in a switch (on+off), so this heat must be minimized.