then your cadence is too high right?

on more than one occasion I've seen this and looked at the cadence of these riders and thought, boy, that sure is a bit fast and I thought they certainly would not be bouncing up and down if they up-shifted slightly. this was even more apparent when I saw 2 recent riders together. one seemed to have an appropriate cadence but his riding companion seemed like he was struggling.

I'm not talking about hip rocking due to saddle height, too high, I'm talking about a rider who is literally bouncing up and down in his saddle with each pedal stroke.

Chicken and egg.

If you are bouncing, it means you haven't trained at that cadence to not bounce. You may be able to drop your cadence to eliminate it, or you could maintain the cadence and train your pedal stroke and core to not bounce.

I ride a single speed, so shifting is not an option. If I want to spin at 130 rpm, I have to train at that and learn to contain the bounce.

Assuming your bike is properly fit to you. I see you mentioned saddle height but theres alittle more to it.

If you're bouncing up and down, it means your form sucks.(setting aside fit). Good sprinters can spin 140 rpm smoothly.

High cadence is good for sprinting, accelerating, responding to attacks.

Or you really have to poop.

When I worked as a courier, I rode fixed 98% of the time, often without realistic breaks. Sometimes had a tight butt due to more than the 350km a week.

Training is also true, riding fixed certainly helped for that.

If you're bouncing it means you're pushing down on the pedals when you shouldn't be. Pulling back at the bottom of the pedal stroke instead of pushing down will fix most bouncing. It just takes a little practice, usually at a higher cadence, not lower.

Finally some usable advice...although I'm sure someone will contradict it soon enough...

If things are set up correctly bouncing stems from poor technique... you have to train for those high rpm efforts and be able to spin those pedals with buttery smoothness.

It can be commented that the OP's form isn't good...but begs the question...how do you develop a better pedal stroke where you can turn higher RPM without bouncing. I believe greg offered good advice. I think about this when I ride in fact. I try to work on my pedal stroke and it has paid dividends in speed and endurance I believe.

Just pushing down isn't a good formula for a pedal stroke OP.
A couple of visuals are:
-Like scraping the mud off the bottom of your shoes at 6 o'clock.
- Pedal in circles.

If you think about it, pushing down hard on right side, can be inhibited by too much weight on the backside of left foot between 6 and 12 o'clock which opposes pushing down on the opposite side because the crank arms are connected of course. If you think about spinning both feet tangential to the chainring, then there is less chop between power zone and dead zone and hence less bounce.
This for me at least creates more natural power with less tension or force in my legs. More effortless speed than focus on just pushing down harder. Focus on pushing down without thought to pedaling in circles inhibits the upstroke which subtracts power from the down stroke in the power zone on the opposite side.

I find I need to use ankling when I get over 120-130 rpm to smooth out the stroke. Downhill on fixed gear will expose weakness in this area. Ankling may be a crutch but I find it helps.

Simplest way to pedal fast without bouncing is not to push on the bottom of your shoe. Pedal entirely with the upper. Keep your feet flat and wiggle your toes. At first you won't be able to do that for more than a very little while, and it's not a way to pedal really, it's just a drill. But a very useful drill.

I almost never bounce, but when I am hammering in the saddle sometimes I start to float off because I am droping the hammer... dialing it up to 500W.

If you have a proper fit: Form is the issue.

The way to learn to pedal at a high rpm is to pedal at high rpm.

Fast pedal drills help with this: Easy gear, 120 rpm for 2 minutes, repeat 5 times. Work up until you do 10 minutes with no difficulty.

One legged drills also help with developing a smooth pedal stroke.