There is a misconception out there that shelter dogs are more prone to be sick or have more medical problems than purebred dogs. This is just not true!
While responsible breeding can mitigate purebred medical issues to a degree, you must do your research, and be prepared that these pups can be very expensive. Make sure you are not buying from a profit-motivated seller less concerned with the long-term health of their puppies' bloodline.
Second, you could buy a purebred puppy that was born in a puppy mill with little to no regard to health problems or safe conditions (which the puppies require to thrive). Puppy mills have been routinely proven to have much lower standards of care, resulting in preventable sickness and injury, among a myriad of other abuses.
Third, although there is not a ton of data either way on the subject, there is some evidence to suggest that mixed breed dogs have some health advantages, particularly concerning several congenital health problems that often do not show up until two years of age or more.
Fourth, although regulation is often left to individual shelters, the fact is that due to pet overpopulation, shelters are sometimes forced to select animals for euthanasia. Known medical problems make dogs much less adoptable. Often out of a need to make room or due to budgetary restraints, such special needs dogs are the first selected for euthanasia.
If a dog was relinquished for health problems by their owner, and the shelter decides to give that dog a chance, then they will share that information with you to make sure that you are prepared to handle the time and cost investment of a known medical problem. Once again, shelter staff are concerned with finding the right forever home for the dogs in their care.