All of us are bombarded with radiation from all kinds of sources, 24 hours a day, and will be,all of our lives. We are exposed to it when we eat certain foods - nuts, bananas, red meat and potatoes all contain tiny traces of radiation.
We evolved to tolerate the amounts of radiation coming in from the sun, and from the planet itself. Nobody really knows how much we can tolerate without being harmed, and it's useless to speculate. The amount you get from sitting in front of a flat screen monitor is tiny - you'll get more from the back of it.
If you're experiencing discomfort when you sit in front of the screen, move it further back,get a different monitor, or use a filter. There's not much more to be said.
As far as hospital X-rays are concerned - the risk varies according to which part of you is exposed. An X-ray of your chest, teeth, arms or feet is the equivalent of a few days' worth of background radiation. That rises to the equivalent of few weeks if your neck or skull is involved, and for breasts,hips,spine, abdomen,or pelvis the exposure is equivalent to a few months or a year's worth of background exposure, depending on the precise location. The biggest equivalent exposure comes from a Barium meal - that equals a few years' worth.
By and large the risk from having an X-ray is far lower than the risk of not having one, and allowing a problem to go undetected.