Does it contradict special relativity that an electron beam in a television picture tube can move across the screen faster than the speed of light?

This is an example of what is sometimes called the "Marquee Effect." Think of the light bulbs surrounding an old-fashioned movie theater marquee, where the light bulbs turn on in sequence to produce the illusion, from a distance, of a light source which is moving around the the marquee.

There is no limit on how short the time interval is between one light turning on and the next turning on, so the perceived light source position can move arbitrarily fast, but in fact nothing is actually moving at all.

In the case of the television screen, the phosphors on the screen can be lit in rapid sequence, but the electrons in the beam do not ever need to move at (or even near) the speed of light.

More generally, there are loads of examples of some imaginary or conceptual "object" moving faster than light, but in all these cases there is nothing actually moving at all. A classic example is the intersection point of two nearly parallel lines, which moves very rapidly as the angle between the lines changes. In this case it is obvious that the moving "object" isn't moving at all, but its still a good example of a case where you can discuss something moving faster than light without there being any violation of physical law.

This is a conflation of phase velocity, and group velocity. The beam can be seen to move from say left to right at higher than c, but no information or particles are traveling that fast. Information is being transmitted from the electron gun to the phosphor at well under the speed of light.

It has nothing to do with the media it is embedded in. The information is going from the electron gun to the screen, not from one location on the screen to another.