> A modern dishwasher will use 3 to 4 gallons on a run. By comparison, my kitchen sink holds about 10 gallons of water on each side. When I wash by hand, I'll fill one side with soapy water and rinse each dish individually. Easily more than 10 gallons of water get used in the whole process.
I'm pro-dishwasher, but you could use much less water handwashing.
If I don't have a dishwasher, my normal method is to stopper one side of my sink, squirt some dish soap on the first few dishes, and run just enough water to wet the dishes. Then I scrub some dishes, run the water (into the stoppered sink) just to rinse them as I transfer to the dish rack, then turn off the water and repeat. The dirtiest dishes that have the most food stuck on get done last so they get the most time soaking in the soapy rinse water from the rest of the dishes. I can do a full dishwasher load with one side of my sink maybe 1/4 full of water.
Time how long you run the sink while washing and rinsing. If you run it for more than 1.5 to 2 minutes, you've used more water than the dishwasher would have.
I'm collecting all the water in the sink, I can measure the volume directly. 10 cm of water in my sink in about 13 litres. My dishwasher is specced for 16.5 - 29.7 litres on the "Energy Saver" cycle that I normally use.
(The "normal" cycle is specced for 11.0-27.7 litres but uses more electricity, which is more expensive than water.)