You describe an effect called "beam blooming," which is a serious problem for high-current beams of low-energy charged particles. These beams are electrically charged and their own space charge causes them to spread outward or "bloom" in flight. To avoid this problem, a TV picture tube uses a low-current beam of high-energy electrons. Because there are relatively few electrons in the beam, they are far apart and the energy stored in their repulsive interactions is quite small. And because each electron is given an enormous amount of energy (typically 25,000 electron-volts), the beam is very stiff-it takes some pretty severe repulsive interactions to deflect its electrons significantly to the side. The tube is therefore able to focus its electrons quite tightly and illuminate tiny spots on the image screen.
Answered by Lou A. Bloomfield of the University of Virginia