After I light it, my gas water heater pilot light always goes out when I release the pilot light knob. This occurs regardless of how long I have been depressing the knob (usually several minutes) to keep the pilot light going after I have ignited it.

Faulty thermocoupling. Replace. Available at Home Depot and other big box stores.

Yes, one of the experts here has this great drawing they often post, but it sounds exactly like the termocoupling