Dips also hurt my elbow,
I don't have a popping sound "which would mean that your tendon is moving out of place during the mouvement" but I have my elbow that shocks me, sometimes I have to stop doing them because of the pain.
What I found aleviate the issue is putting an elbow brace : something like this "attention This one is very expensive in my opinion"
It will put pressure on the tendon therefore you'll have less risk of it popping out.
I also watched a few videos in which they advised to work the shoulders as it my be due to a bad alignement with the shoulder that the elbow tendon pops...
And some advice to warmup!
at the end everybody's different. But yeah, get yourself an elbow brace! it should help a lot!