I've been trying pretty hard to make a workable design for a plane that can turn very quickly, but all of them seem to result in missing checkpoints a lot.

Do I add more wings? Decrease the length? Decrease weight? What do I do?

The obvious answer is to decrease weight as much as possible.  Try using the flourescent tubes found from climbing L.O.G.'s tower to connect critical vehicle components, doing that always gives me a chuckle.  Another thing I've learned is that you can use the suck and blows for forward propulsion as if they were jets you could activate, deactivate, and reverse.  Failing that, you can always just build a hoverpack by slapping a seat and some fuel onto a small jet.

I broke my months long posting hiatus to offer you these words of guidance, do not let them be for naught. 

Adding more wings has seemed to help me, but that probably increases the weight a bit which is what SolemnOaf said not to do. Personally, I've found that the best way to not miss checkpoints is to just not give the vehicle full throttle. Just focus on little short bursts and slow down enough to make turns. I have no idea what tubes he's talking about. One of the first things I did in the game was climb that tower and I don't recall unlocking any tubes.

you must use the tubeses, it is your only hope. 

But seriously, you didn't see the crate up there that had the Mr. Pants statue and the Flowerpot and the Windshield and stuff?  You might want to go back, none of it's really useful except the flourescents cause they're 3 blocks long and weigh next to nothing. 

Another thing I thought of is you could try using folding propellers to turn your plane into a helicopter for extra mobility whenever you need it.  Don't forget about aerodynamics when building boats or planes, either.  Having a nose or at least a bevelled underside helps.

I have all the other stuff, so I guess I have them. I probably just assumed they were merely cosmetic accessories and didn't bother trying to build with them. I'll have to keep that in mind next time I'm playing it.

Just for the heck of it, I thought I'd mention that I built a pretty sick hang-glider-launching vehicle. I modelled the vehicle after a dragster, so it's totally loaded with engines and power but definitely intended for flat straight runways. Anyway, at the front of the vehicle, I have a detacher gadget connected to only a seat and two folding wings. So anyway, with this thing, I get as much speed as possible, then detach my seat and open my wings and man, that thing travels such far distances without even having an engine. It doesn't serve much of a purpose in missions, but it's fun to just glide around on and works well for moving from mission to mission in levels like Nutty Acres.

I was hoping that the Gyroscope would fix this problem, but it's too tricky.

"The obvious answer is to decrease weight as much as possible.  Try using the flourescent tubes found from climbing L.O.G.'s tower to connect critical vehicle components, doing that always gives me a chuckle.  Another thing I've learned is that you can use the suck and blows for forward propulsion as if they were jets you could activate, deactivate, and reverse.  Failing that, you can always just build a hoverpack by slapping a seat and some fuel onto a small jet.

I broke my months long posting hiatus to offer you these words of guidance, do not let them be for naught. "

Power your plane with two Suck'n'Blows.  That cuts out the weight of engines and fuel.  Also is surprisingly efficient.