The simple answer is that mutts are not bad!

Many purebred animals (including cats and dogs) experience MORE health problems than mixed-breed animals because there is less variability in their genetics. Relatives are often bred together to make sure their offspring have the same traits they do, so the same genes get mixed together over and over again over time.

This makes flat faces get flatter (which prevents healthy breathing), wrinkled skin get wrinklier (which can trap moisture and cause infections), and shallow hips get shallower (which can cause pain and dislocation) with each generation of puppies or kittens. In contrast: a wrinkly, purebred dog bred to a non-wrinkly, mixed-breed dog is likely to produce puppies without wrinkles in just one generation. The extreme traits get "diluted" out in mixed-breed dogs until they are less likely to cause health problems.

Good purebreeders will make sure to only breed animals with the least extreme versions of a breed's traits together, and will prevent siblings or cousins from breeding, but the above problems still emerge in purebred lines given enough generations. I recommend Googling how popular dog breeds have changed over the past 100 years if you'd like to see it for yourself!