Can You Just Get On An Untrained Horse And Ride It?

It's a favored western trope. The cowboy ropes the wild horse, puts a saddle on it, gets on and rides off into the sunset.

How realistic is that?

With a horse that has never been handled - not very. In reality, when cowboys did that, they would have to deal with the animal's violent objections, which often involved somebody getting hurt.

With a domestic horse that has a calmer temperament, has been treated well and handled her entire life, then it actually is possible. Colored cobs (what Americans call gypsy vanners) and some lines of Quarter Horse are known for being "born broke." I personally know a mare who's first ride was a random teenager and who's second ride was a total beginner (She was so good her owner did not know she was not trained).

Most horses, however, will show some degree of annoyance when saddled and ridden for the first time, ranging from trying to walk away from the tack to violent bucking. So, if you want to use this trope?

With a wild horse, make sure you talk about how difficult a task it is and how the horse needs to be "bucked out" and honestly...wild horse breaking generally involves letting the horse be mad until it's too tired to keep fighting and behaves.

With a domestic horse, make mention that the animal has a good temperament, or is from "good lines" that are known for being easy to train.