*disclaimer: I am not a western rider. I ride huntseat and dressage* However, that being said I've worked a lot on sitting my butt down in the saddle for dressage. My trainer often tells me to really think about someone pushing down on my shoulders. I don't know why, but that visual really helps me. If you are pinching with your thighs, it will make you bounce more. In fact, you're probably also subconsciously pinching at the knee, which essentially creates a hinge and makes you bounce. You need to think about elongating your leg, wrapping it around the horse and moving your hips with the horse. A lot of sitting with no stirrups on a kind, patient horse (particularly on the lunge line) will help with sitting as well. My trainer also has us do lunge lessons in which we remove our stirrups and practice a number of exercises. I definitely won't do a good job of explaining them but I'll try.
1. Take your arms and point them straight at the ground, hand flat and fingers pointing at the ground. Think about reaching for the ground. Helps deepen seat and improve posture (which in turn will deepen your seat).
2. Reach down with one arm (don't lean forward) and grab your ankle on the same side as the arm you're using. Bring ankle up towards hip and ride this way, holding ankle. Improves seat and posture.
3. Hold onto front of saddle, bring knees together and have them touch in front of you while riding. Deepens seat.
4. Ride with a dollar (or another lightweight object that won't spook your horse) under your butt. If it falls out, you're doing it wrong.