You can't even call the Essex-class super carriers, either. They were of somewhat heavier displacement but not much bigger in size or airgroup than Enterprise after her 1943 refit. The Essex-class weren't even the largest carriers of the war. Shinano out-displaced both by almost double, but she was intended for use as a support ship, not a true fleet carrier.
As was said, the Forrestal-class was the first of what we would define as a "super carrier." Midway was the first American vessel to be given the appelation.