You need more practice, then more, then a little more after that. The fear goes away as you get better at doing it.
You're right to start out small, with quiet streets at quiet times. That gives you the time you need to get a feel for operating the vehicle: How it accelerates and brakes, how tightly it turns, etc. You need to know that stuff before you take the next step, so take the time to learn it.
The next step is an actual trip somewhere and back. Go to the local store or drive-thru, then come back home. Don't do it at rush hour and don't go downtown, and definitely no three-hour cross-state trips just yet, you're still keeping it small. You're entering into live traffic, but you're staying on familiar ground close to home.
After a few of those trips, you move it up a notch and do a cross-town or almost-downtown run. The traffic will be heavier and the stress will be higher, but you're already used to steering and working the pedals so you'll be able to handle it.
See how it works? You increase the difficulty level gradually, just like when you do anything else. Before you know it, you'll be bombing down the expressways or twisting through crowded downtown and cursing at all the idiots with the best of them. But take the time you need to learn how to cope with it, and here are some tips you can use any time you want...
- When you make a mistake, and you will, take a deep breath and let it go. Remember, there is no such thing as almost having an accident. You either get into an accident or successfully avoid one. Almost crashing means you successfully avoided a crash. When you feel that panic urge rising, take a deep breath, hold it for a second, let it out, and remember that you successfully avoided an accident. Try to remember not to make that mistake again, but don't hold it against yourself.
- When other people make mistakes, and you can be sure they will, do the same thing. You'll feel a sudden urge to jump out and beat the living snot out of them, but don't give in. The other guy wasn't trying to be an idiot, wasn't trying to cause a crash, he just made a mistake like everyone else. Deep breath, and let it go. When you're driving a motor vehicle, which is easily one of the deadliest weapons on earth, panic and rage are not your friends.
- Controlled Breathing: Sometimes you'll find yourself in a lousy situation you didn't plan to be in. You intended to take a nice easy trip, and you end up lost in heavy bumper-to-bumper traffic. And it's getting dark, and you're not sure how to get home. And a storm just blew in, so now you can barely see. You'll feel a really strong urge to freak out, but do this instead. Nice slow deep breath in (natural breath, don't stuff your lungs), hold it for one second, then let (don't blow) it out. Wait one second, and repeat. Controlling you breathing will help you stay focused and control your actions. This technique sounds dumb, but it's used by astronauts in space, firefighters in burning buildings, and race drivers on the track. If it works for them, and it does, it will work for you.
- The Break: This one is really important and really helpful, and not just for new drivers. I've been driving over 25 years, professionally at times, and I've seen the wildest conditions and heaviest traffic known to man. And sometimes, it just gets too much for me. The break is when you pull off the road, step out of the vehicle, and give yourself a rest. You can do it anywhere any time, it's always there if you need it. Have a coffee, stretch your legs, smoke a cigarette (even though smoking is bad), look at the pretty flowers, whatever. Just get out of the driver seat, take a zen moment, and you'll get your center back. Ten minutes, half an hour, whatever. Even if you're trying to get somewhere on a deadline, use The Break when you need it. Because getting there on time is no good to you if you're dead.