I've been cycling in traffic for 30 years and never been hit by a car once. I've cycled in Ireland and England where you have to share narrow roads with cars, buses, and 40-foot trucks. I now live in the San Francisco Bay Area and I bike to work just about every day, and the infrastructure here is far more bike-friendly than in the UK.
In the morning I ride on the expressways. Yes. Expressways. Non-cyclists think it's dangerous, but it's actually safer than regular surface streets since there's a big wide shoulder to use as a bike lane, and there's less cross traffic with no private driveways. It's also easier to get up to my cruising speed that averages 17 to 18mph, which is a lot faster than the cars stuck in bumper-to-bumper traffic and going nowhere. I always make a mental note of which cars I see when I get onto the expressway because I generally get to my destination either before them or at the same time as them.
In the evening I take a longer route home which is mostly on trails and only a few miles on surface streets. The trails are like superhighways for bikes since they pass under so many of the surface streets and skip dozens of stoplights.
Cycling is perfectly safe as long as you know what you're doing. It's no more dangerous to ride a bike on a road full of cars than it is for a car to drive on a freeway full of trucks.