The things to remember for every one of the S License tests are:
1) exit speed
2) stay off the walls
3) exit speed
4) learn the right braking points
5) exit speed
6) throttle control in certain areas
7) exit speed
If you look at the demos, the car is very twitchy because the demos were made without an analog controller. You can do a much more smooth run and beat the bronze time by a few seconds if you just avoid the walls and use the right amount of throttle through the turns.
For example, look at S-2 (GT40 at Seattle Circuit II). You hug the right, then just barely tap the brake and cut hard left through the first chicane. Then you hug the right again, then use hard full brake just before the rumble strip, cut hard left and left off the gas for a split second, then full throttle again. Then you go into the box turns. You veer left over the first rumble strip, then let off the gas and you make big right turn around the box. As you're going into that last box, you hit the brake a little, then just hold the accelerator half-way down as you approach the railroad tracks turning hard left, then accelerate out onto the backstretch at full throttle. Good exit speed will gain you a lot of time on the straight. I guess I won't go over every turn, but there are a lot of places where I let off the gas completely.
S-1 (Speedster at Apricot Hill) is pretty easy. Cut way inside on the first left, then slow down to about 67-69 MPH for the next left. If you hit that 2nd turn at the right speed, you don't have to slow down all the way through the S curves, and you don't even have to slow down for the huge left. At the hairpin at the end of the straightaway, you have to slow way down to about 50 MPH and cut way inside. The important part is hitting that last chicane between the cones right. If you can get through there at 68-70 MPH, you'll probably get gold.
S-5 (Trueno at SS5 Wet) isn't that bad. It's all about throttle control. You have to slow down a lot for that hairpin in the top-right of the map, then you can stay on the gas all the way till you go down the hill on the backstretch. Then you have to slow way down for the hairpin. The last part is the hardest, and involves very good throttle control, or else a lot of tapping the accelerator to get through with a good line.
S-6 (Viper at Laguna Seca) is pretty tough. You just have to get good at Laguna Seca. Slow down to about 50 MPH as you enter the first hairpin, then gradually get on the gas as you come out of it. The Viper will lose control if you just slam the throttle all the way down while you're turning, but you need to use a little throttle to get out of the turns with a good speed. You don't have to slow down much for the next right turn if you have a good line, and you barely have to tap the brake at all for the next right (if you had a low exit speed on the first right, you actually don't have to brake at all for the next right). The most important things are: (1) Slow down to 80 MPH and cut in simultaneously between the #2 and #1 braking points on the first big left. (2) You don't have to slow down at all for the big sweeping left at the top of the map. Keep on full throttle and cut hard left a little after you pass the #3 braking point. (3) Learn to master the corkscrew. Aim far to the right side as you approach the skidmarks, then hit full brakes right before the dark skid marks as you come over the hill. Stay outside! Don't cut to the left until about when you get to that #1 marker, then as soon as you get turned most of the way into the corkscrew, get on the gas just a little -- about 1/3 to 1/2 throttle -- and just drive down to the inside rumble strip on the right. As soon as you're pointed the right direction to get out, you can use full throttle. You only have to let off the accelerator to make the next left (you don't need to brake if you have a good line). (4) On that last sharp left before the finish, hug the right and use full brakes when you pass the #3 marker, then make sure you have a smooth exit.
Laguna Seca can be a harsh lesson in throttle-lift oversteer. You actually have better traction when you're pushing the accelerator slightly through a turn. You will spin the car if you let completely off the gas through the corkscrew, unless you slow down way too much.
S-7: Even if you don't know how to drive a manual, use a manual tramsission on this test. Don't shift below 2nd the whole time. You'll be taking almost every turn in 2nd gear with the gas half-way down. If you understand this concept, you should be able to get silver on this test in your first attempt. I actually bumped the left wall before the final stretch, ruining my exit speed, and still got gold.
S-8: Same as S-7 (except even more important here), you should use a manual transmission. Do not shift below 3rd. You will be in 3rd gear almost the whole time, which will keep you from spinning in the tight turns. You have to shift up for the big straights, but just don't go below 3rd and this test is ridiculously simple.