Is it normal to fill up the grey water tank this fast?

I sympathize with you! We had the same experience the first few times we went camping. We learned to economize: the two of us now use 5 gallons a day, total, for showers every night and dishes and bathroom. I wash dishes with a trickle of water, using easily rinsed dilute Castile liquid soap (Dr. Bronners). And the showers are Navy style -- get wet, water off, soap, and rinse carefully.