A shower is generally considered to conserve more water and the energy that it takes to heat up the water. But in fact is depends upon the size of the tub and the flow rate of the shower head. Most tubs provide for a typical bath capacity of about 35 gallons.
Shower heads are rated from a low of .5 gallons per minute (gpm) to a high of about 7.5 gpm. Laws in the US require shower heads to give a flow rate of less than 2.5 gpm. The water pressure also plays a factor in the final determination of the gpm.
You can test your shower head by capturing the water that flows from it for one minute. (or 30 seconds and double it...) The question then becomes how long are you going to be in that shower. 3 minutes?
5 minutes? Or are you not coming out until you solve all your problems. At 2.5 gpm it would take (35/2.5=) 14 minutes to equal the 35 gallons that might be found in a typical bath.
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