I live on a ranch and at times in my long ago past I have spent many hours on horseback. I also have very large balls at least from what I have seen as comparison.
Supportive underwear and tighter than average pants are a must, as is the positioning of your genitals as you dress. The gait of the animal makes a very large difference in how you ride it. Good Buddy, there is no exercise better than horseback riding, and I strongly urge you to adapt to the inconveniences because over the long term you will come to love it. 87 Sultan has said some great things, hungcowboy also gave some great advice. There are differences not only between individual horses but between the breed. I have had a great deal of time on the backs of drafts and draft crosses. These have (to simplify it) a "different shape" than many other breeds and require a different saddle. I have heard it compared to riding a 55 gallon oil drum, but as I have not ridden any oil drums I would not know. One of the most beneficial things you can do is to take your camcorder or digicam and have some friend take some movies of you riding if possible from the side.
Look at what you are doing in relation to body position, spinal alignment, and saddle position. The other thing I strongly endorse if you decide that you want to work to fix this, and that is your own saddle that is a proper fit for you. Many novice riders do not know that saddles are "sized" for the rider and if one is shall we say "off by miles" it can take a really pleasurable experience and turn it into a exercise in discomfort even for an experienced rider if it is more than a very short ride. To me personally in riding there is a time of adjustment. During this time, you learn to become one with the horse instead of just existing as dead weight in the saddle. One of the hazards of this time period is the examination of your own sanity for wanting to do it. Over time, as cowboy mentioned, your muscles change and react by instinct to what you should be doing instead of "after the fact". Even with all that, I have said in other postings that I am the victim of a bucking where I sustained some pretty heavy duty injuries. The accident was a total fluke and I do not fault the horse to this day, if one wanted to call it an act of God that would be the closest one could get. The animal bucked when it was hit by a wind blown chunk Mesquite bush that hit his left flank thorns and all. The saddle was a roping saddle and I came down on the saddle horn. Again, this is my only problem and I am now not very far from 60 years old. Right now you can see you have some support and some people will help you through the "rough times" on this. Keep with it and a few lessons probably wold not hurt either.
The one thing I can tell you is that a true relationship with a mount is unlike anything else you will ever have. Keep at it and over time you will definitely get it right and really will enjoy it.