I agree with zeugnimod in that you're not using enough of the track. I think this is your biggest problem. You need to learn to use the full width of the track on the entry and the exit of the corner. By using most of the track (taking a wide line into and out of the corners) you're basically making the corner more open, so you can carry much more speed at all points (corner entry, apex and corner exit). I don't know how your understanding is of these terms, so I'll give a little explanation of them below. If you already know, feel free to skip them.
Corner entry - By definition this would be the window between when you first turn the wheel to enter the corner and when you hit the apex. It's logical to include the braking in this though, as braking will have a massive impact on your corner entry. Braking can be an art in itself, so I've added an entry on that at the end. On the actual turn in, try to be smooth. If you turn in too aggressively you'll upset the car, which will generally cause it to have increased understeer or oversteer depending on the setup. At this point in your driver development you're looking to have the car neutral on the way into the corner - you don't want either the front or the rear to be sliding more than the other.
Apex - This is the point at which you stop entering the corner and start exiting it. It's just a transition point between the 2 stages.
Corner exit - This is the point where you're coming out of the corner so you have to start accelerating again. You should be unwinding the steering lock as you exit the corner, so there will be a time when you're steering at the same time as accelerating. At this point it can be easy to spin the driven wheels and end up facing the wrong way. This is a big problem in the FO8!
Braking - We all know what braking is but there's an art to doing it well. The goal is to decrease your speed (and gear) until you're at a point where you're going at the correct speed in the right gear for the corner. The correct speed and gear for each corner will vary widely based on car and conditions, this is mostly down to practise. Try to be smooth under braking, don't lock wheels for long periods of time and change down the gears smoothly - space them out a bit. If you change down too quickly you'll overrev the engine, causing it damage and potentially locking the driven wheels.
Now, onto more specific advice for your driving. Use more space! Look at the replay I've attached and check out how much of the track I'm using - I'm all over the track and running over the kerbs too. By doing that, I'm able to carry a lot more speed than I would be able to by using half the track. In the attached replay I've taken my time to get up to speed but I've always tried to use all of the track. You should do this too. Start off at a comfortable pace (it doesn't have to be quick but you have to be in control at all times!) and try to maximise the amount of track you use. Once you're happy you're using enough, start going faster. Brake a couple of metres later, turn in to the corner with a bit more speed and try to get on the gas a couple of metres earlier. Don't try to make big jumps. At this stage you're looking more for consistency than outright speed. Drive 100 laps in 10 lap stints. In each of those stints, try to put in at least 5 laps within 1 second.
Once you're comfortable doing that, start pushing harder again and repeat the process.
You're going to end up making mistakes (locking brakes a lot, crashing, spinning and running off the track). Don't be discouraged by this, it happens to everyone (even F1 drivers do it...). The important thing is that you should be making less mistakes when you start doing more laps, if you find yourself making the same number of mistakes, slow down a bit for a couple of laps then try again.
I'm not sure how much you'll be able to take from this post, but I hope it helps in some way. I've attached a replay and a setup that may help.
Good luck and have fun learning!