The key here is that you think there is no skidding. In fact, there is skidding, although for normal automobiles this is barely noticeable. For normal cars, the rear wheels simply skid a lot less than would the front wheels when a turn would be fully forced.
You can see this also in trucks, where it becomes necessary to have dual or triple-axle steering when doing tight turns while manoeuvring.
The answer has to do with the form of static friction.
There are two types of friction, static and kinetic. Static friction is the type of friction that occurs when the bit of object that's touching the ground doesn't move (relative to the ground). Kinetic is the type that occurs when there is relative motion. Let's focus on static friction.
Consider a block sitting on a level, rough table. There will not be a frictional force. Why? Well, frictional forces are parallel to the surface (i.e., horizontal with the block on a rough table scenario). If there were a such horizontal force with nothing else acting on it horizontally, the block would be accelerating on its own. But we don't see this.
Now, magnitude of static friction forces can vary; it can range from zero up to some maximum value. The maximum value is $f_\text{max}=\mu_s N$, where $\mu_s$ is just a number that depends on the two materials in contact (e.g., wood block on plastic table), and $N$ is the upward normal force from the table. In order to start to move an object from rest, you have to exceed the magnitude of the static frictional force $\mu_s N$. When this starts to happen, the object will begin to move and kinetic friction will take over.
When you try to gently lift the block upward, but you don't actually let it leave contact with the table, the normal force $N$ decreases. This is because your upward force plus the normal force from the table has to counteract the downward gravitational force. If you apply a large upward force, the normal force doesn't have to be as large.
With this decrease in $N$ comes a decrease in $\mu_s N$, and hence a decrease in the maximum static frictional force. This means the force you need to apply in order to overcome $f_\text{max}$ is now less if you apply a small upward force to the block.
That's about it! Note that this is a very "macro" explanation that hinges on the form $f_\text{max}=\mu_s N$, which is just a very good experimental relationship. There are probably more microscopic explanations that could be more appealing to you.
Best Answer
That depends entirely on you. You are the author of the puzzle. As others have pointed out in comments, if the tubojet is pointed downward, and if its thrust exceeds the weight of the vehicle, then you don't need the wall (or the car's engine) at all. The turbojet alone can lift the car.
If you choose to point the jet horizontally, so that it only presses the car's wheels against the wall, then you need to meet two conditions for the car to be able to climb:
The car's regular engine must be able to generate enough torque at the wheels to lift the weight of the car, and
The jet engine must press the wheels firmly enough against the wall that the static friction between wheels and wall can hold the weight of the car.
In order to know how much thrust the jet must make in this case, you need to know the coefficient of friction between the rubber tires and whatever it is that the wall is made of. The greater the coefficient, the less hard the jet needs to thrust.
According to my sources, the coefficient of friction between rubber tires and a concrete surface is somewhere in the neighborhood of 1.0. If that were true, then the thrust provided by the jet would have to be the same as the weight of the car.
The most economical answer in that case would be to point the jet straight down, and not use the car's regular engine at all.
If the coefficient of friction were greater than 1.0, then you could maybe find a more economical solution using less thrust, at a different angle, but I'll leave that as an exercise for the reader.