Here is what I did (I had a hell of a time trying to complete the dirt and snow, road was easy because i used the RX-8 concept race car too)
1) Spent $750,000 on the peugeot rally car
2) Went driving in the snow, it sucked BADLY (too light?)
3) Sold the peueot race car
4) Bought the Suberu Sti Spec c '05? (White one, not the rally)
5) Decked it out to get about 500kW, no weight reductions (buy the tyres)
6) I turned off ASM, TSC can stay on but i turned it off as well
7) In the snow events (ice arena ESPECIALLY), accelerate completely to the corners and just before you come to the corner either...
a) Use handbrake to spin rear around so that the rear of the car bumps into the outer edge of corner (stopping you VERY quickly) then accelerate out.
b) Turn car towards outer edge of corner (you end up being right up against the wall, cant turn into the wall any further) and just use the outer edge to guide you around the corner
NOTE: Methods (a) and (b) are usefull in different circumstances, its up to you to decide which method works better.
I also find that ramming the car you are racing is usefull sometimes as well, but nudging your way in between without the penalty is much better
methods (a) and (b) are also sort of cheating, and so is the ramming. So if you dont like cheating it isnt the way to go (I myself hate cheating, but i also hate going so damn slow in the snow events)
good luck!