With young drivers, I did read/hear somewhere that the thing to do is look hard at the choice of car.
Many have mentioned small hatchbacks - which on the face of it are relatively cheap, but on the other hand they are popular amongst the young. Ergo, they are the cars that get crashed spectacularly on a depressingly regular basis. Which of course makes insurance companies jumpy.
The gist of the article, or whatever it was, was for the young driver to go a bit off piste with his/her choice of rolling iron. For example, look at getting a small to middle size saloon car with a sensible engine size which is a few years old. Random example - VW Bora might turn out to be cheaper than a newer Polo... From memory, the writer managed to insure a BMW 3 series saloon for less than a Corsa, or summat...