I started going on bike tours when I was 13. I lived in Southern California at the time, there was lots of traffic, but it never bothered me or the other riders. Over the course of hundreds of group rides and many tours, I have never seen an accident involving a rider and a car, though I have seen countless crashes between riders. However, in nearly every case, no one was hurt enough that they couldn't continue the ride.
One good thing about cycling (at least when I started) is that cyclists had a little more class than people who practiced many other sports. Drugs, alcohol abuse, and other problems were quite rare among cyclists. To many riders, cycling has it's own drug-like effects, without the dangerous health side effects, cost, and risk of arrest.
A person your age is safer on a bicycle than behind the wheel of a car.