The key thing is not to make such a big deal about about cheating publicly. No public witchhunts, no Turnitin submissions, etc.
Include your plagiarism/citation policy clearly on your course syllabus, and briefly go over it on the first day of class, but that's it. Don't have students submit through Turnitin directly, but use Turnitin yourself discretely.
Depending on the severity of the infraction, the first one would result in a stern warning in private, or no credit on the assignment. Subsequent infractions warrant more severe punishment.
If you let them off "easy" in their eyes (but not too easy) you're the good guy, but they'll still know you're serious about it.
On the other hand, if you announce you're using Turnitin, or otherwise publicly dwell on cheating, then students will feel antagonized and band together against you. If you don't give them a reason to discuss it with each other they won't. As you mentioned before, the social pressure among peers to not cheat is strong.
But by ignoring the cheaters you're doing both them and their non-cheating peers, and the university as a whole, a disservice.
Kids straight out of high school might not even know what they're doing wrong, and if no one ever tells them they'll grow up to be cheaters in life too.
Of course it's also unfair to the non-cheaters who work harder and receive the same grades, and ultimately the same degree.
On a larger scale, if a university tolerates widespread cheating the quality of their graduates will decline, which obviously isn't good for anyone.
[This is a repost of my comment on the blog]