@Roger V.'s answer is a good one and I've upvoted it. There are some other additional plausible reasons worth adding:
First (and probably most importantly), mankind evolved in the Old World and this gave diseases many more opportunities to jump to humans. From what we can tell, human diseases mostly are zoonotic in origin. Covid-19 and SARS almost certainly came from animal diseases and jumped to humans in the last fifty years. (This is independent of whether Covid-19 was a lab escape, since if it was, it was from a lab studying animal diseases which might jump to humans!) Green Monkey disease, Marburg virus, and Ebola all appear to have jumped from non-human primates fairly recently. AIDS itself probably has an non-human primate origin. We regularly get influenza strains from birds, but where it evolved first is, as far as I know, unknown -- birds may have caught it from us, originally! Mad Cow disease made the jump to humans less than fifty years ago.
Bottom line: Humans in the Old World lived for longer (it's no accident that Africa is a great source of new diseases) and with more distinct pools of wild animals (giving more opportunities for diseases to evolve) than the people of the New World.
Secondly, the Old World seems to have had greater movement of populations, providing the dubious benefit of regular waves of epidemic disease spreading around. Even as a percentage of the population, many more people died in the Old World than in the New: Bubonic Plague killed half of Europe. The Roman Empire during the reign of Marcus Aurelius suffered from a plague -- unknown which -- which was nearly as deadly. Seventy-five years later the Plague of Cyprian was nearly as deadly. It has been plausibly argued that both came back to Europe with returning armies. And there were many more.
The people of the Old World's "advantage" was that the entire population had been killed off several times over, but not all at once. People in the New World probably got more than one disease at the same time and suffered the same losses all together.
Finally, it's important to remember that people and diseases will in time come to an accommodation. The course of Covid-19 is a good example: When Covid first hit, its effect was pretty serious, but even then it affected immunologically naive old people far more seriously than the immunologically naive young. Today, there are no immunologically naive people of any age left: There are very, very few people who have not either been immunized or gotten Covid or both, and the effect of the disease is greatly muted.
But besides that, Covid itself has mutated to become more contagious. It's done so by preferentially lodging in the upper respiratory system which makes it easier from virus particles to become airborne and go off to conquer new worlds. But the deadliest effect of Covid was infections deep in the lungs, so this evolution -- entirely to Covid's reproductive advantage! -- has muted the danger it poses to humans.
The SARS outbreak in 2003 only infected a few thousand people, but there is some evidence that those people had some natural immunity to Covid (a close relative of SARS) when it came around. And Covid has probably armed all of us with at least some immunity to the next SARS-family virus to make the jump to people. Living in a bigger stew of diseases, Old World people carried both diseases and acquired immunity to those diseases that people in the New World lacked.
This is a grossly over-simplified summary of a large topic, but I think it covers the main points adequately. Two very readable books that go into more detail is the classic Plagues and People by William McNeil and the more recent Guns, Germs and Steel by Jared Diamond.