
A car's suspension system, simplified, is simply a series of giant springs that help even the car out on rough surfaces, and try to neutralize or deaden some of the G-forces felt by the driver. An analogy is a waterbed. If you push on one end of the waterbed with a sharp motion (say a punch), the effect will ripple out slowly to the rest of the waterbed. Same thing happens with shocks. If you hit a rock on your right front tire, rather than the whole car bouncing up, you merely feel a slight jolt. Depending on the size of the rock of course.

This has other effects too, which become more pronounced at higher speeds. If you accelerate suddenly from stop, if the grip of your car is good enough, rather than peeling out (burning rubber), you'll feel the weight of your car shifted towards the rear wheels. This is all Newton's first law. Something you should have learned in primary school. If you brake firmly, but smoothly so as not to skid out, you'll feel the weight of your car transferred towards the front. Now, with cornering at high velocities, it's the same thing. The car's weight will shift (due to centripetal force) to the opposite side of the direction you're turning in. If you're turning right, the car will shift left. Left, it shifts right.
This was rather comically portrayed in Desert, where at medium speeds, the whole car would actually rock up on two wheels. It's much more realistically done in TM2. However, that, coupled with the lack of advanced air control seen with the previous titles, means that if you have any curve input upon leaving your jump, the car will torque as is seen in all the videos. The reason for this torquing? It's in the air. With a lack of repulsive force and counter-input (the ground and being able to steer on the ground), the car's inertia simply carries it in the direction it was last traveling, in this case, forward due to acceleration, left or right due to however the driver was turning, and around the Z axis because of the laws of motion and the car's suspension.
Now, once you've read all that, come back and tell me it's still a bug.

EDIT: On your second video, that also explains why he's pressing the keys with little tap, taps, rather than actual presses. He's trying to minimize the effect I outlined above.