I don't know how you begin to solve this.
I doubt it has anything to do with the pitch clock. The TJs in the first 100 days thing SportsGuy posted doesn't suggest the pitch clock has anything to do with it, nor does MLB's research showing guys who work faster aren't at higher risk for TJ.
As long as there are millions of dollars at stake for throwing hard, people are going to throw hard.
Maybe MLB could implement some kind of mandatory pitch count, but then guys will throw even harder since they don't have to last as long in the game. Maybe MLB can put a limit on how fast a pitch is allowed to be. Something like if the pitch is over 95 MPH it's automatically a ball, but that's not fair to the guys who can throw at high velocities easier, and I'm sure the MLBPA won't stand for it.
I think the only way this gets better is if technology improves to the point where we can get more out of arms without destroying the elbows. And even then, people will still push the limits to outcompete their peers.