I've noticed while cycling on my road bike that, when I try to pedal very quickly, I start to 'bounce' a little in my seat. As in I'm lifting off the seat a little with each pedal, so feel I'm not harnessing the full strength of my legs.
To avoid this, do I need to adjust my bike seat/handlebars? Or is this more relating to good cycling form? Or, perhaps, I just need to go up a gear at this point?
Most likely, you should just simply switch to a higher gear. A needlessly high RPM is just wasting energy moving your legs up and down. This energy is not usefully utilized to propel the bicycle forwards.
However, do check your seat height. Either too low or too high seat can cause problems.
I had a 5-year break from cycling. Now when I started cycling again (I have only cycled 100km after the break), I noticed my pedaling RPM is significantly lower than it used to be. If I try to pedal at high speeds, I feel I am unable to produce the needed power without rapidly tiring my legs. I suspect my optimal RPM will go up as my muscles get used to cycling again after the break.
This legs-rapidly-tiring phenomenon is different from "bouncing". If your legs are not strong enough, you feel you are just unable to produce useful power at high RPMs. You don't "bounce".
So, your "bouncing" means that you are either using so ridiculously high pedaling speed that no active cyclist would use, or either your seat height needs to be adjusted.