A bunch of important things to do with flight of any kind scale linearly with atmosphere density... dynamic pressure is an important one (which in turn affects lift and drag).
This means that if you were able to double the density of your atmosphere, all sorts of things get twice as easy... flapping might develop twice the force, and wings can be smaller or gliding can be more efficient, etc etc. Unfortunately it also means that more force is required to push your way through the thicker atmosphere... there's no such thing as a free lunch.
All the problems of aerodynamics will still apply, and horses aren't really quite as streamlined as birds are even before you put a lumpy human with a giant draggy cross-section on their back. Classical dragons look a wee bit more streamlined to me, which makes them seem like the better choice for fast movement in the sky. You should consider using mythical creatures that are much more birdlike as your fast mounts... maybe something more like a hippogriff than a pegasus? Still not sure how to tuck those drag-increasing long legs away, though... horses just aren't made for flight!
I suspect the Quora question was specifically about the atmosphere of Titan. The thing about Titan is that yes, the density of the air is high, but the gravity is also much lower: about a seventh of that on Earth. Without significantly reducing gravity as well as increasing density it'll be tricky to get your mounted pegasi off the ground.
There are other problems with warm, thick atmospheres on low-gravity worlds (eg. everything blows off into space) but that's another question.
could a pegasus be light enough to fly and still strong enough to carry a person even in such an atmosphere?
Maybe. Clearly, human-powered flight exists in the real world, and humans and horses have pretty similar constraints on muscle density and power output. Throw in some avian adaptations to improve streamlining and reduce mass and you could just handwave things in as being plausible. I still think you'd need to reduce gravity a little, though... not necessarily as much as Titan, but down a bit from Earth-normal.
Passengers might need to be dressed in special streamlining gear, and use a special harness to stay attached.
Flying on something that is pegasus sized but shaped more like a bird than a horse is likely to work much better. A little simurg might be more suited to your needs. Could probably eat a pegasus, too.