Dinosaurs would be much less popular, dragons it depends on if dragons are still alive
There's a saying in paleontology: if turtles were entirely extinct today people would be fascinated by their remains. Turtles are bizarre, their bodies are so strange with their limbs inside their ribcage, skulls that make no sense compared to living reptiles, that if turtles were extinct today it would be hard for anyone to imagine how they could exist. Now imagine a turtle that is sixty feet long, has wings, and breathes fire. There's your answer for dragons.
If dragons were real dinosaurs would almost certainly be less popular, as dragons are basically dinosaurs+. Part of what makes dinosaurs enticing to a lot of people is that they were once real, and if dragons took that uniqueness away there isn't a lot of reason to favor dinosaurs over them. Dinosaur paleontology would be a niche subject, kind of like how mammal paleontology or paleontology of Triassic archosaurs is today, with the possible exception of a big area of interest in theropod dinosaurs that evolved into birds (but with far fewer people liking dinosaurs for being big and toothy). Think of how the "less interesting" Mesozoic megareptiles (mosasaurs, ichthyosaurs, pterosaurs, plesiosaurs) are treated today. Always showing up as side roles in documentaries and movies about prehistoric life, almost never the stars of the show, and almost always less popular or more likely to be inaccurate because there isn't as much interest in them.
If dragons were alive today there would be even more interest in them and even less in dinosaurs, as fossil dragons would be seen as important in understanding the origins of dragon biodiversity. Something similar happened with dinosaur paleontology, in the early 20th century dinosaurs were seen as big, dumb, uninteresting evolutionary dead ends that were good for attracting children but unimportant as far as science went, and instead a lot of research focus and popularization was placed on fossil mammals because fossil mammals gave rise to modern horses, cats, dogs, humans, etc. Nowadays things are reversed because of the Dinosaur Renaissance and the dino-bird connection, which has revitalized dinosaurs as active animals and evolutionarily important because they gave rise to modern birds and aren't just an evolutionary side-show.
Another aspect is that before the discovery of dinosaurs dragons didn't look that much like dinosaurs, they looked like mashups of snakes, big cats, hooved mammals, eagles, bats, and crocodiles. It was only after dinosaurs became popular that dragons started looking like dinosaurs with bat wings. If dragons were a real animal it doesn't take much of a leap to think that people would take features of real dragons to make new mythological monsters that are seen as even more alien and fascinating than we find dragons.