From the Super Mario Wiki
He is supposed to be taking Missy and Bud to Wild Waldo's One-of-a-Kind Amusement Park by a long road trip, but ends up driving into a Warp Zone that accidentally leads the family into the Mushroom World. Norman, distracted by the strange scenery, assumes it to be Iowa and he accidentally crashes into two walls of Blocks. The car almost gets totaled and Norman inspects the steaming engine, while a Mushroomer, who is fishing, goes screaming "ALIENS!" because of the strange vehicle, people, and the smoke coming out of the engine.
After a while later, when the Mushroomer is spreading the rumors, Norman jumps to the conclusion that he is in Nebraska, not Iowa. Mario and Luigi soon save the station wagon from the Doomship, but not the magnet that the Doomship sends out to abduct the station wagon.
King Koopa and his Koopalings are skeptical of Norman's family and Koopa thinks that the aliens are trying to disguise themselves to take over the Mushroom World. Norman is scared and he threatens to report "Waldo" (as Norman's wife call Koopa) to the traveling agent. But then, Kooky von Koopa then takes Norman's family to his new invention, the Transmutational Brainwave Analyzer, to gain their thoughts about taking over the Mushroom World.
Norman gets really freaked out and he says, "I just got enough of this, this is not fun anymore!" Just after he says that, the Mario Bros. come and rescue his family and then offer the family to take them to the Warp Zone where they came from. Norman gets confused and asks Mario what is going on and who is he.
Norman then drives out of the Warp Zone back into the real world where they came from. Then, while Norman is driving, Missy asks if they are there yet and Norman just slaps his face as a response.
In the Super Mario World episode "Rock TV," Koopa spies on a real world family in a flashback. Among them is a man resembling Norman, except he has balding grey hair, a plain green shirt and no sunglasses.