Ever heard of hybrid vigor? That's a good thing. I think mixed breed animals tend to thrive as opposed to purebreds due to limited genepool as well as physical requirements. Most animals are bred for a look, not a function so the purebreds tend to faulter from it. I'm talking about all animals too.
It really depends. If the breed of purebred has a limited gene pool and/or do not come from a good breeder who also breeds for health and function beyond the latest trends then they are more likely to have problems than mixes. If the purebreds come from a good breeder with a good gene pool they are less likely to have problems than mixes. Same possibilities with the mixed breeds. You have just as much chance of offspring being born from a mix that have all the bad genetics from both breeds as the good ones but usually people with mixes aren't breeding for a specific trait. Instead they are just taking the healthiest animals that survive the easiest and letting the rest go because that's all they are interested in. If the same was applied to a purebred with only keeping the ones that survive the best instead of breeding for specific traits that don't help the breed then you would have just as hardy of purebreds. Unfortunately humans tend to interfere too much and breed for traits that aren't helpful while ignoring the bad and including weak individuals in their breeding programs just because they have the color or look that is desireable.
I got a purebred dog from a good breeder in order to avoid genetic and health problems and to have a healthier longer lived animal. But I spent a year looking for a good breeder and researching. Some people specifically get mutts because they have only dealt with poor breeders or don't know how to find a good one so they've only seen weaker purebreds than mutts. It really comes down to the breeding program and how responsible the breeder is not just the specific breed or mix.