Sharks and humans belong on very different branches of the tree of life so it is basically impossible for sharks to catch infection diseases (germs like bacteria or viruses) that have evolved to infect humans.
This happens because, to start an infection, the germ must interact with molecules on the outside surface of cells in the infected animal (human or shark). This interaction is very intimate and depends on the shape of the molecules. So a virus that has the right shape to stick to a human cell will be unlikely to stick to a shark cell. It’s like trying to fit a piece from one jigsaw puzzle into a different puzzle – it won’t work. So a human virus can’t infect a shark
There is also some speculation that sharks don’t get cancer. I’m not sure if this is true or not but Wikipedia seems to doubt it.
I would guess that sharks are unaffected by many of the diseases which affect us. Stephen will know more about this, but I think Infectious diseases (spread by e.g., bacteria and viruses) are usually adapted to one or a few different hosts, and can’t inflect very different creatures. A shark being a fish is very different from a human being a mammal. However big jumps can occur (e.g., flu moving from birds to mammals). Non-infectious diseases often occur when something goes wrong with the normal functioning of the body. For this to affect both fish and humans it would have to be some aspect of normal functioning which is more or less the same for both of us – so maybe something to do with basic metabolism (i.e., the chemical reactions which make our bodies work – some are very similar even in very different creatures).