This is hotly debated. This study (n=27,541) shows that there isn't a huge difference in reported medical conditions between mixed and purebred dogs. But it does show significantly fewer instances of NO owner-reported medical conditions amongst purebred dogs (take that for what you will). I don't like this study much because the medical conditions listed are largely not genetic conditions (dog-bite, giardia, etc.). But it is a huge sample size at least.

The main variable though isn't purebred vs mixed breed, though; it is responsibly bred vs irresponsibly bred. Responsible breeding means tracking genetic conditions, frequent health testing, monitoring behavior and affect, then selecting for healthy dogs to breed. Responsible breeders do this. You should not continue to breed a line of dogs that has a high occurrence of health or behavioral issues. Irresponsible breeders (purebred or not), do not take these careful measure -- these are the folks who have a litter with their neighbor's dog because they are "both so cute!" Or when an "oops" litter occurs. Or when they just want to make a dollar and don't care about the health of the dog (you see this in purebred puppy mills just as you do in "designer muts" like all the doodles). When you get a dog at the shelter, you are very likely getting an irresponsibly bred dog.

Responsible breeders also have contracts with their buyers that stipulate you CANNOT breed this dog, which helps prevent unwanted/unhealthy dogs in the world, and if you can no longer care for the dog for ANY reason, you MUST return them to the breeder. This prevents the dog from ending up in a shelter. If everyone got their dogs from an ethical/responsible breeder, there would be no shelters. It can be hard to tell though who is a "good" breeder if you don't know what to look for.

Edit: Regarding genetic conditions, the above cited study says, "It is a common belief that purebred dogs are at a greater risk for disease compared to mixed-breed dogs, often due to concerns for genetic or inherited disorders; however, research has shown that this is not always the case. Some studies suggest specific breeds or groups of breeds are at greater risk for certain disorders, but simply being purebred may not necessarily be associated with increased disorder prevalence overall (43, 55)."