How do I fix slow water flow in my refrigerator door dispenser?
If the ice maker if refilling quickly after making a batch of ice, then the problem will likely be in the valve in the door dispenser or some kind of blockage in the line leading there.

The first easy fix that comes to mind is an ice buildup, since I've seen that wreck havoc on in door dispensers. You can try turning off the fridge and taking a hair dryer to the dispenser from each side to defrost any buildup, and then do your best to dry out anything that drips.

Beyond that, this will probably require taking the dispenser apart, possibly replacing the valve, or it may be as easy as cleaning the spout.