Why does having a large bowel movement make my right testicle hurt?

Increased pressure: It shouldn't hurt. When you have a bowel movement, it causes pressure in the abdomen and pelvis to evacuate the stool. This pressure can be transmitted to your testicle and be sensed as pain. It's worth getting a testicular exam to make sure nothing else is going on.

Large bm: Testicular discomfort that only occurs after passing a large stool is more likely muscular strain type discomfort in the muscles between the rectum and scrotum(perineum). To be sure, you could have a physician or urologist examine your testicles as well as check you for an inguinal hernia which also becomes symptomatic after straining.