Your steering servo just isn't up to the task. I switched from a Hitech 5645 to a Futaba 9351 on steering and it made an amazing difference in the car's steering.
That Hitech that they are talking about is an awesome servo, I know lots of guys that run them and have never heard one single complaint, and never heard of one breaking.
For steering you need a servo that is both fast AND strong... that means spending a little more money. Sorry, that's racing!