No water pressure in bath tub after turning off hot water heater

I have a leak in my hot water heater. Haven't repaired it yet, but I have shutoff the input (cold) water line going to the water heater.

I have closed all the hot water shut off valves on my sinks. The sinks have plenty of cold water coming out, but my bath tubs have very little water coming out. Both tubs have that style of faucet that you turn the knob counter-clockwise to get hot water and clockwise for cold, which will eventually turn off the water.

Is the water back flowing into the hot water supply? If so, where the heck is it going!? I don't notice any more water leaking out of the hot water output supply line.. Is there an easier explanation/fix for no water pressure in the bath tub?

Any ideas?