Pretty much just by definition. The supernatural is that which is outside of nature. Nature is equivalent to the universe. If something is outside of the universe, it cannot interact with the universe. If it could interact with the universe, it would be part of the universe. If it cannot interact with the universe, there is no meaningful sense in which it can be said to exist. At best, you could say that it exists in some other universe*, but not this one.
Note: this is not intended to apply to any specific "supernatural" phenomena. This in no way disproves god or ghosts, or whatever. It merely says that if they do exist, they are not supernatural.
*Though, by these definitions, other universes don't exist. They either interact with this universe, in which case they are part of this universe, or they don't, in which case they don't exist. I'm fine with this, since "exists" should imply existing in this universe. It's not really much good to talk about something that exists, but has absolutely no impact on us in any way whatsoever.