Odd datasheet, by "starting current" they mean "stall current". 4A is the full load
current at nominal full power.
In general you don't need to provide the full stall current, especially if you ramp up
speed with PWM. Expect the power supply voltage to dip at start up if it can't
provide full stall current, this is to be expected (not every power supply can deal
gracefully with this though).
For more than 1.5A you have to go to discrete MOSFET bridges (see note below though)
as single chip bridges cannot carry the current without overheating.
There are what look like single-chip H-bridges rated at 20A or more, but they
are actually several MOSFETs and a control chip mounted in a single package,
such as this: