Why do you want to turn the sod over? Right side up is probably better and a lot less work. Do you have access to cardboard? I would mark out the area for the garden, put down your kitchen scraps, prunings, and shredded paper. Wet all this well, then lay on the cardboard. Make sure it is wet -- you can even soak it in a garbarge can (Dave's idea) You can mulch over the cardboard if you want it to look better, otherwise its not necessary.
To plant, just make a hole through to your original soil. The cardboard will smother the grass out. If you use unshredded paper -- such as newspaper just make sure its is wet.
