Your console has an issue interacting with the DSP chip in Super Mario Kart. If you have any other DSP games like Pilotwings, I guarantee you that they will also suffer.
Years ago I had similar issues with this game also, but only in one of two SNES consoles. As soon as the race started, the graphics would go screwy and the game was unplayable.
The cart worked fine in another console, so your comment sounds very likely.