General elbow pain is often associated with an imbalance between the forearm flexors and extensors. In most cases (but not all), the forearm flexor is in a state of overuse because of repeated "palm towards elbow" patterns, and you end up with an inflamed elbow tendon.
This is one reason why I recommend a thumbless grip for all pressing movements. If you're using a thumb-around grip for every movement, then the forearm flexor is repeatedly contracting without dispersing that tension across the forearm extensor. The top of the forearm is just hanging out up there, not really doing a whole lot.
So for exercises like lat pulldowns and rows, use straps and don't wrap the thumb. For pressing, a thumbless grip tends to disperse the tension and torque a bit more evenly throughout the wrist, elbow, and shoulder joint, along with not asking the forearm flexor to bear the brunt of the work.
Now include some direct forearm work, along with using a thumbless grip as often as possible in other movements. For the forearms, one-arm cable reverse curls and some high-rep hammer curls work very well. I'm talking 30-40 reps per set.
After a month of this, you'll most likely be pain free if that imbalance between the forearm muscles is the root cause of the elbow pain. Then you're free to smash arms a few times a week and rid your life of sad little biceps.