Seem like you would need 3 tables and perhaps 4 depending on whether or not you want to have specific show info. Artist, Exhibit, Show, Show Exhibit.
Show Exhibit would store the showID and exhibitID for each exhibit in that show.
Thanks! Currently we only have one table. I don't have enough experience with relational data. Is there another (simpler) way to get around it?
I am hesitant to suggest but this may work for you...
try a repeating field (allows for multiple values for one field)
If you plan on continuing to develop additional features for this database, this will limit you greatly, if not, it may be the simple solution you are looking for.
In many ways, the multiple table approach recommended by Mr. Vodka, is truly the "simpler approach". Other options, like a repeating field, may be simpler to set up, but then prove much more difficult to work with.