Take a look at ZeroDesolation's solution above. This is essentially what got me through the time trials.
However, there are a couple of things I would like to add. There are no penalties for using assists (traction, stability, ABS, etc.). So, if you need them, you should be using them! I went ahead and turned on ABS, traction control, and (sometimes) stability control. Additionally, if the race line is off or on "braking only", turn it on to "full". The racing line helps immensely, and there's little reason not to use it here (no penalty!). Although the brake points (yellow-red sections) aren't 100% efficient (you can often brake less or later), at least you'll have a good race line to follow in general.
Because many of the cars in Time Trials are fast, powerful, and unfamiliar, it takes a lot more concentration and skill to beat the listed time without assists. TCS (traction control) cuts power to the engine when your tires lose traction (usually during acceleration), preventing you from going nowhere or skidding into a wall. ESC (stability control) applies the brakes to individual wheels during a skid, helping you to keep going in the right direction. ABS (antilock brakes) maximize braking effectiveness by automatically hitting the brakes in rapid succession to prevent a long lockup. Not only does this increase the effectiveness of braking, but it also prevents an out of control skid that may result from locking the brakes. The racing line provides an "optimal" line that would give you the fastest time possible. As I said above, it isn't perfect, but since there is no penalty for using it (or any other assist), you might as well use it!
Essentially, the Time Trials are a set of hot laps that push you to become proficient at driving many different cars on many different tracks. While following ZeroDesolation's solution and by using all the help you can get, these time trials become much easier. Keep your foot on the gas (or finger on the trigger?), hit the corners hard, stay on the track, and drive your way to victory!