I love cycling (I cycle a lot), but unfortunately this isn't true. In the US, you are roughly twice as likely to die as in a car[1]. It is much, much, much safer than motorcycling! (Somewhat old data, though)
If you ride your bike 3 times a day, every day for 70 years, you have about a 1.6% chance of dying. This is not unduly dangerous.
Injuries for cyclists thus are far more likely. Even collisions at lower speed that basically leave a car dented and the occupants uninjured will result in injuries for cyclists: The classic dooring accident, run over by a turning car, etc. Even overtaking the cyclist with too little safety distance can lead to a fatal accident [1]
The stats in Berlin (2015) indicate that while cyclists are involved in about 4% of all accidents, they make up 21% of persons killed (pedestrians are even worse off: 1.33% involvement, but roughly 40% of all persons killed). They also make up more than 30% of all injured (30% of major injuries). Pedestrians make up for 13% of the injuries (24% of major injuries). So the group of soft targets that is involved in roughly 5% of accidents suffers 60% of all fatalities, 54% of nmajor injuries and 43% of all injuries.[2]
[1] see incident 10 on this list http://adfc-berlin.de/radverkehr/sicherheit/aktionen/62-geis... (sorry, german only) [2] data taken from https://www.berlin.de/polizei/aufgaben/verkehrssicherheit/ve... (sorry, german only)
Having said that, there are a couple of things to keep in mind. First, you want to be predictable. That generally means following all the traffic rules. When riding in stop and go traffic, keep your place and don't pass on the right. When turning left, take the whole lane (or walk your bike across the intersection if you are nervous to do that). Realise that you have really good power to weight ratio, so there is no reason at all not to use a whole lane at low speeds. Don't ride on sidewalks. Don't dart in and out of parked cars, etc. Keep your gears in good condition and make sure you are in an appropriate gear to accelerate when you need to.
The biggest danger in a city is riding beside large vehicles (like trucks or busses). Avoid riding beside them, even in multi-lane situations -- either ride in front, or slow down and ride behind them. Keep enough space between you and them that they can see you in the mirror (there is a blind spot directly behind the vehicle). Normal cars are not a problem, it's just vehicles that make wide turns.
The next biggest problem is people opening doors when exiting a parked car. Generally speaking, take a lane when passing a parked car. If you can't do that (because of traffic), slow down to the point where you can stop if required.
After that, the worst thing is usually stupid drivers that overtake you and immediately make a right hand turn. Not much you can do about that, so be sure that you are ready to brake when you get to intersections or entrance ways. Make sure your brakes are in good condition and learn how to brake effectively (front brake is where almost all your braking power comes from, shift your weight back to stop from going over your handle bars). You can brake considerably faster than a car, so it's not actually that dangerous if you are paying attention.
Most important is to practice in order to get your confidence up. Make a very short circuit that you can ride every day. Maybe walk it a few times first so that you are super comfortable with it. Ride at whatever speed you are comfortable with and don't feel bad if you want to pull over and take a break. It won't take you long until you can try riding in other places.