first step is going in the steam controller setting in the steam overlay and switching your gamepad to actually act as a gamepad , steam consider this game to not support gamepad and will usually set the controller to act as a keyboard.

then I recommend setting the dpad as arrow key , start and select as enter and escape so you always have an easy way to nagivate the menu reguardless of what buttons you set on the controllers.

once this is done you can go in the game itself , enter the options and select your controller , depending on the resolution the game is set at it is possible that you get issues where the controller name is not displayed properly but pressing right or left still switch the device.

then lastly you can enter the configuration menu and set the keys the way you like , setting the controllers properly will ensure that you actually have analogue controls while turning.

here is some images to help better.

it is possible that the game interpret your controller triggers as inverted.

like if you set nitro on Right trigger and handbrank on left trigger when you go in the game itself
RT will handbrake and LT will push the nitro.

so if your set the gas and brake on the trigger and it doesn't seem to work make sure that its not working in a inverted way.