ok just played it and finished it first time, i have two tips one is a bit better than the other.
1. Keep a steady hand, i tend to play with two hands holding the controller.
2. You are allowed to touch the sides! they will slow you down but as long as you make it in under 20 seconds you still pass the stage.
That being said i just did it in 19 sec and touched the sides twice right near the end so i think it could be done without touching the sides at all.
oh just noticed you said "you have to be quick and not touch the sides to have any chance.. " so you might know that you can touch the sides but it is faster not to.
My only other tip then would be that don't take the shortcuts, i usually go under the first round mound at the top right of the course then over the top of the second one and then i never take the shortcut at the end as the wide corner is pretty easy to get around quickly. Although i have seen a mate go straight through the shortcut without touching the sides.