Your Sword Coast druid probably hasn't seen a dino
Sword Coast Adventurer's Guide doesn't mention dinosaurs at all. That's not decisive as it's not an exhaustive setting book, but it is the closest thing to a "pure" setting book for the Realms that's been published so far for 5e.
Xanathar's Guide to Everything gives some hints as to where druids might see dinosaurs: the plesiosaurus appears on the Coast and Underwater tables; the pteranodon, allosaurus, anklyosaurus, and triceratops on the Grassland table; pteranodon on the Mountain table. So maybe you've seen a plesiosaurus on a 4th-grade whale-watching trip, or a menagerie traveling up from Calimshan had an allosaurus with it? (The Coastal Encounters tables include pteranodons and plesiosaurus later in that book. Grassland Encounters include anklyosaurus, allosaurus, pteranodons, triceratops, and even tyrannosaurus rex.)
However, there's nothing in 5e's literature of the Sword Coast (SCAG, Tyranny of Dragons, Princes of the Apocalypse, Storm King's Thunder, and other adventures) to suggest that dinosaurs are native to that part of Faerûn.
Hop a boat south, though…
…And you'll arrive in Chult, and Chult is teeming with dinosaurs. You can't spit without hitting one! Really, just take a look at this Tomb of Annihilation promotional piece that depicts Chult's Port Nyanzaru: