No, it’s not true. Sharks actually get bacterial infections, viral infections, and they also suffer from parasites, in short, the entire array of infectious diseases from which all the animals and plants must protect themselves. Sharks also get cancer, as opposed to another disseminated myth.
The claim that sharks do not get disease at all or suffer from cancer is mostly spread by advertisements that sell shark cartilage as a dietary supplement. These supplements always claim to prevent disease or cure cancer, but there have been no rigorous medical tests to protect people from disease.
The immune system of sharks is very much like ours.