Here are some facts:
In 2013, 32,719 people were killed in road accidents.
Also in 2013, 4,735 pedestrians were killed. The leading cause of injury to pedestrians is tripping and falling, but by far the leading cause of death is being hit by a car.
For cyclists in 2013, 743 were killed. The leading cause of cyclist deaths is, again, being hit by a car.
At this point, even though these numbers show that bicycle deaths account for only .02% of deaths in the US each year (which completely disqualifies cycling accidents from being anywhere close to a health epidemic) people will inevitably point out that not many people ride bicycles, so that 743 is actually quite a large number. Of course cars kill 20 times as many pedestrians as they do cyclists, because there are so many pedestrians, and so few cyclists, right?
While there's nothing wrong with that statement, the fallacy is in thinking that you can just extrapolate from there with simple multiplication. Cycling accounts for approximately 1% of all trips taken in the US. So the logic is that if cycling accounted for 100% of all trips, instead of 32,719 we'd have 74,300 people dying in road accidents! Terrifying!
Except it doesn't work that way. Remember that the leading cause of death to both pedestrians and cyclists alike is cars. So if 100% of road traffic were cyclists, there wouldn't be any cars to kill anyone.
Now is a good time to bring up The Netherlands, a country where cycling accounts for 27% of all trips, and in cities (where accidents of all types are much more likely) that number rises to more like 50%. Even worse, practically nobody wears bike helmets there! So clearly, their hospitals must be overflowing with people suffering from exploded skulls. One can only imagine the carnage on their streets... because it doesn't exist. In 2013, 184 cyclists died in The Netherlands, which is .001% of their population. Our 743 account for .0002% of our population, but if you multiply that by 27 (1% vs. 27% of all trips taken) you get .006%, which is 6 times as many deaths per capita.
In other words, it's much safer to be a cyclist in a country where nobody wears a helmet but everybody bikes, than it is to be a cyclist in a country where virtually nobody does it, but at least you have a piece of styrofoam strapped to your head.
Why? Because when more people bike, we build more infrastructure to support bikes. Drivers become more aware of bikes, and instead of just gnashing their teeth and dangerously swerving around them, they're familiar with how bikes move through traffic and see them as just another part of driving on a road. Cyclists are more visible, motorists are more prepared, and everybody gets along better. And bikeshare programs are exactly the kind of thing to promote more people riding bikes.
But wait, we still haven't trotted out everyone's favorite last-ditch argument when all else fails! "No matter what you say about big-picture statistics, you'd still be safer wearing a helmet."
Yes. This is true. Adding more shock absorption to the human skull can protect the brain from traumatic injury. In fact, it's so true that it applies to virtually every human activity. So let's really follow through, and make it mandatory to wear a helmet while driving or riding in an automobile. Let's make it mandatory for pedestrians to wear helmets. Why stop there? Let's make it mandatory to wear full body armor any time you leave the house, and let's make it illegal to sell or build anything with sharp corners. Let's spend our entire lives suspended in smooth carbon fiber tanks full of Jell-O, so that nothing is a danger to our well-being until a heart attack ends it all!
Incidentally, heart disease is the leading cause of death in the US, is a real health epidemic, and is preventable by putting your butt in a bike seat and riding a few miles every day.
Sources:
http://www.pedbikeinfo.org/data/factsheet_crash.cfm
https://en.wikipedia.org/wiki/List_of_motor_vehicle_deaths_i...
https://bicycledutch.wordpress.com/2014/04/25/traffic-fatali...
https://en.wikipedia.org/wiki/Cycling_in_the_Netherlands
http://www.treehugger.com/bikes/why-dutch-dont-wear-helmets....