Right now, I can't think of any way for a graph to require $5$ colors yet not have a $K_5$ minor ($5$ vertices all connected to one another). If there is indeed no such graph, then every graph that requires $5$ colors must have a $K_5$ minor, which by Kuratowski's theorem means that these graphs are non-planar. Thus, every planar graph is at least 4-colorable.
I highly doubt the proof of the four color theorem is this simple, so could someone show me a counterexample to my argument?
See the Wikipedia article on the Hadwiger conjecture. The conjecture states (as a special case) that a $5$-colorable graph must contain a $K_5$ minor. This fact has never been proven directly, however it was shown to be equivalent to the four-color theorem. Since the four-color theorem is true, we know the Hadwiger conjecture is true for five colors, and so no graph of the type you describe exists.