You shouldn't bounce, but sit down into the gait. You should not be constantly standing on the stirrups, as it were. You are riding poorly. Obviously, the millions of male riders down the millennia did not have your problem, so they could ride without pain.

Now, perhaps you think the Persian Immortals in Herodotus or the cavalry of Xenophon just somehow tolerated the agony because they had never known the comfort of stirrups. So let's step up to the 20th century, rather than my boring you senseless with all the stirrupless riding traditions (Sioux and Cheyenne, Rome's cavalry, fully armoured Renaissance gallowglasses).

Spanish Riding School of Vienna: Much of the early training is done without stirrups, to develop a good seat. If you check a YouTube vid of them, notice that the "airs above the ground" are done w/o stirrups.

Not to mention everyone on a long trail who takes his feet out of the stirrups to rest his legs, or rides bareback for the fun of it. IIRC, the movie Gallipoli has a bareback scene, galloping in the surf, and no actors were injured.

Sources:

Alois Podhaijsky, The Training of Horse and Rider (he saved the school and the horses at the end of WWII)

Hi! Not only am I /r/AskHistorians Equine History commentator, but I'm also an avid equestrian and riding instructor.

Riding without stirrups is not terribly difficult, provided that one knows how to sit properly.

Xenophon wrote this in his manual on horsemanship:

But now, supposing the rider fairly seated, whether bareback or on a saddle-cloth, a good seat is not that of a man seated on a chair, but rather the pose of a man standing upright with his legs apart. In this way he will be able to hold on to the horse more firmly by his thighs; and this erect attitude will enable him to hurl a javelin or to strike a blow from horseback, if occasion calls, with more vigorous effect. The leg and foot should hang loosely from the knee; by keeping the leg stiff, the rider is apt to have it broken in collision with some obstacle; whereas a flexible leg (9) will yield to the impact, and at the same time not shift the thigh from its position. The rider should also accustom the whole of his body above the hips to be as supple as possible; for thus he will enlarge his scope of action, and in case of a tug or shove be less liable to be unseated. Next, when the rider is seated, he must, in the first place, teach his horse to stand quiet, until he has drawn his skirts from under him, if need be, (10) and got the reins an equal length and grasped his spear in the handiest fashion; and, in the next place, he should keep his left arm close to his side. This position will give the rider absolute ease and freedom, (11) and his hand the firmest hold.

Basically, he's saying that you should sit as if you were standing up straight. This means that you should not be sitting too far forward or too far backward on your your pelvis. The position is similar to that of a person walking with good posture. When you walk, you typically don't crush your genitalia betwixt your legs, and if you do, you have much bigger problems.

Now, for what you are actually sitting on. Here is what Mark. W. Cross & Co (a professional saddle maker) had to say back in 1891:

What is a man to sit on? Well, he has two bones in his seat, which we venture, in imitation of German phraseology, to call his "sitting bones."

Basically, to sit properly on a horse, it was accepted that one should sit on a triangle consisting of the seat bones and the tailbone.

Cross goes on to explain some of the differences in sitting positions (he also plagiarizes heavily from Francis Dwyer). But basically, you should be on your "sitting bones" or "seat bones." (I reference Cross and not Dwyer because Cross is more accessible to the layman than Dwyer. However, Dwyer's work is clearly superior in depth and understanding of the topic, and was written prior to Cross').

The next important part comes from having a relaxed and supple seat. A rider with a relaxed seat can move with the back of the horse in such a way that they don't bounce. This involves relaxing the pelvis and learning how to balance on the back of the horse properly. Imagine a mild swing forward and backward with your sitting bones. Alternatively, pull out a wooden chair without armrests, sit on the very edge of the seat, and with just your hips, try lifting the back legs off the ground. This is similar to the motion needed to properly sit on a horse.

Now, this requires the rider to have the necessary body control to maintain proper posture and leg position while riding, which is not an easy thing to do. This is why one does not see large numbers of shock cavalry prior to the invention of the stirrup- basically, it was hard learning to swing a weapon from the back of the horse. The stirrup also facilitated leaning from side to side, which is arguably the most important aspect of fighting from the back of a horse (and not, as some have speculated, the ability to brace against the cantle of a saddle while holding a lance).

Your problem was probably that you aren't a terribly experienced rider, and you rocked too far forward on your pelvis. This is a problem I see commonly with beginning riders. They hunch forward in an attempt to find balance, and ultimately rock forward and on to the front of your pelvis. Alternatively, you're arching your back too much, which is throwing your seat forward.

But anyway. Regardless of all of that, it's not terribly difficult to squeeze with your thighs and rise up and off the horse's back. Jumping and galloping bareback is a relatively easy thing to do with a reasonable amount of experience, decent balance, and adequate athleticism.

Personally, I believe that stirrups cause a large number of equitation maladies, especially with beginning riders. This view is supported by the Spanish Riding School, which continues to train riders in the classical way developed in the Renaissance.

Additionally, one of the essays (honestly, I forget which and don't want to reread the entire book) in Karen Raber's book, "The Culture of the Horse" mentions an individual who went to a riding school and learned to ride bareback first, in what the author calls, "the proper way."

Anyway. To make a long post short, good equitation removes genital pain entirely, and people who were brave enough to ride a horse into battle without stirrups or a saddle were typically good enough riders to avoid landing on their junk.

Sources:

Complete Training of Horse and Rider, by Podhaijsky.

How to Ride, by Mark W. Cross and Co. https://books.google.com/books?id=hYIvAAAAYAAJ&lpg=PA19&ots=gXNfbzllnh&dq=man%20riding%20no%20stirrups&pg=PA20#v=onepage&q=barebacked&f=false

Seats and Saddles, by Francis Dwyer https://archive.org/stream/onseatssaddlesbi00dwye#page/n11/mode/2up

On Horsemanship, by Xenophon http://www.gutenberg.org/files/1176/1176-h/1176-h.htm