Turbulence is generally harmless. It's the airborne equivalent of driving over a bumpy road in a car.
The airplane can tolerate much more turbulence than the passengers. In severe turbulence, people who are standing inside the airplane can fall and hit their heads or otherwise be injured—just as a person riding in a car going swiftly over a bumpy road can be tossed around if he's not wearing a seat belt.
That's why you are told to keep your seat belt on while seated during the flight. People have been injured and (rarely) worse in flight by turbulence because they weren't wearing seat belts—it's the most common cause of in-flight injury. Usually flight attendants are the ones who end up injured, because their work requires that they stand a lot of the time. The flight itself is never in danger, though.
Keep your seat belt fastened and you'll be fine. You wouldn't ride down the highway at high speed in a car without wearing seat belts (I hope), so you should take the same precaution in an airplane. The airplane itself won't crash, even if turbulence is severe.
As for wings, they are the strongest parts of the plane. The plane is built around the wings, not the other way around. Anything that can snap the wings will have already destroyed most of the rest of the aircraft. In practice, the only thing that can do this is a thunderstorm … which is why aircraft never, ever fly through thunderstorms. Fortunately thunderstorms tend to be localized, so pilots can easily detour around them. And modern weather radar (on the airplane and on the ground) makes thunderstorms easy to spot from a distance, even at night.
Yes, on extremely rare occasions, turbulence has crashed aircraft, especially small aircraft that don't deal with turbulence as well as big jets. Usually thunderstorms were involved. But this is too improbable to worry about, especially today, since modern aircraft and pilots are unlikely to be taken by surprise by thunderstorms, thanks to all the weather gadgets they have now.