For those finding it difficult to control your car while driving (especially at faster speeds) let me offer some tips which I have picked up. Please keep in mind that I am still new at this game like we all are so these tips may not be the best way and they do take practice to master.
First and foremost, the type of car you are driving makes a HUGE difference. Unlike previous GTA games where the main difference was acceleration and top speed, the difference between the lower end, medium performance, and high performance cars is huge. With that being said, go find yourself a good car to practice with at first. I recommend a Turismo which is a sports car that can be found parked outside of that mafia leader's mansion (Forgot his name. It's Vlad's boss) which is at the southeastern tip of the first island.
Second, you have slow down a lot before making your turns now. Unlike San Andreas where you could haul ass and then use the handbrake to control a tight turn, that unrealistic driving feature no longer exists. Fear not though! With enough mastery, you will understand how to fly through the streets and make proper turns.
Lastly, when it comes to using the handbrake, there is a very different way to use it. The Standard controls have it setup so that the A button is your brake and the left trigger is brake/reverse. Unlike previous GTA games where the handbrake (A button) was enough to handle you car at fast speeds, you now need to use both the handbrake and reverse at the same time. This does not mean you can fly at any speed and then use the combo of brakes and expect perfect results, but it does help a lot.
The basic scenario of what I do for a 90 degree turn:
1. Pick up lots of speed.
2. Start to slow down as I approach my turn by letting go of the gas and tapping the left trigger some (brake/reverse) to reach a speed where I feel comfortable for my upcoming turn. This speed greatly depends on the car I am driving.
3. Hold both the handbrake and brake/reverse while turning and not hitting the gas. I do this for about 40%-60% of my turn.
4. Let go of the brakes and hit the gas.
5. Rinse and repeat.
Again, it takes a lot of practice and I do not even have it down yet especially when it comes to the lower end vehicles. I hope this helps though.