I'd give the edge to Mickelson, but it's a pretty close contest.
Like others, I think Mickelson will win at least another major or two, perhaps even a few more than that, which would truly give him the decided edge over Norman. Most of his career he's had to play alongside Tiger, the dominant player of the era, which no doubt has affected his overall wins and probably his total number of major wins.
Norman didn't face anyone quite like Tiger for the duration of his career, but he did go up against the likes of Jack (very end of major-winning career), Watson (again, toward the end of his major-winning career), Faldo and Seve in their primes, plus the likes of Price and Larry Nelson, all of whom won multiple majors while Norman was contesting for majors. While I think it's fair to say Norman never had to compete against someone the caliber of Tiger, I think the remaining depth of top-tier talent faced by Norman was at least the equal of, if not better, than what Mickelson has had to face.
Another factor on Norman's side which makes this a close debate is that he's clearly shown the ability to compete for titles in all four majors, which really only a small handful of players in the history of the game have done. Mickelson's inability to really effectively figure out the links terrain of the Open Championship is something of a minus on his side. Norman, on the other hand, seriously contended for titles in all four majors, albiet winning only two.
I also think you can reasonably argue both players should have won more majors than they have/did because of their own failings down the stretch of majors.