Aren't X-rays used to 'see' bones more dangerous than UV-C rays from Sun? I am not sure, as to how to go about this ...
But I believe X-Rays are distinguished by their high frequency.
Exposure to a few minutes of UV increases the risk of sunburn. (I do not want to say about X-rays)
Are hospital x-rays safe because of exposure time.. or is it something else
that I can't think of.
All sources I visited only mention that X-Rays aren't dangerous, when they are used in hospitals...
 A: X-rays are indeed dangerous.  However not being able so understand, say, the location and growth of a tumour so you can effectively zap it is also dangerous: in fact it's very often fatal.  So we accept the rather small risk from a CT scan because it is vastly outweighed by the risk of not doing one.  I've had a CT-scan: if I hadn't, and if there had been something bad, I would probably have died – the risk from the CT-scan was something I was extremely happy to take.
The people most at risk from medical X-rays are the people who make them: an individual being X-rayed gets one, or a few doses of radiation in a context where not getting them is likely to have far worse outcomes (see above), while the radiographer making the images may be getting doses all the time.  This is why radiographers often work behind shielding &/or from another room.
The dose required for medical X-rays has also decreased as sensors have improved: when I first went to my current dentist (nearly 30 years ago) he would wear a lead apron to take X-rays of my teeth and they had to have shielding in the walls to avoid people in the rooms next door getting accidental large cumulative doses: he doesn't need to do that any more because the doses have become so tiny.
I don't know whether X-rays are more or less dangerous than solar UV: quantifying that requires understanding the relative doses and how much is absorbed: the human body is rather transparent to X-rays which is why they work for medical imaging, but almost completely opaque to solar UV, so it absorbs almost everything that hits it.
And the Sun is a very powerful source, to which you can be exposed for many hours a day: about 3% of the power from the Sun which reaches the surface is UV (the proportion is far higher, about 10% at the top of the atmosphere).  On a clear day when it is directly overhead the Sun can dump about $1\,\mathrm{kW/m^2}$ onto the surface, which means about $30\,\mathrm{W/m^2}$ of UV.  The average will be far lower than this, but it's doing this all day, while a CT-scan (which I think is the nastiest kind of medical X-ray in terms of dose) takes a few minutes, and you don't have them very often.  Certainly if you spend a lot of time outside in bright sun while not wearing many clothes (and you are pale-skinned) then you're running a really fairly large risk of bad things happening to you.
A: 
Are hospital x-rays safe because of exposure time?

Mostly that. Also, you're rather transparent to X-rays, so many of the X-ray photons pass through you without interacting. But the ones that do interact can certainly do damage. 
Even visible light can do damage, but ionizing radiation like UV & X-rays is more dangerous because those photons have the energy to disrupt the electronic structure of many kinds of molecules.
