Are there any American horse races that fall under the category of "classic races"?

Most Definitely, the Triple Crown Races are Classic History America Races. 

KEY WORD HERE IS CLASSIC-

The Triple Crown is made up of three races; the Kentucky Derby, the Preakness Stakes and the Belmont Stakes. The Triple Crown is not one race by itself. The horse has to win 'all' three races to win the Triple Crown. Triple means three.  The Triple Crown has featured the same three races over the century. The histories and significance of each of these major races is what makes them Classics in American Thoroughbred horse racing.

The Triple Crown races are:

1 The Kentucky Derby- Held at Churchill Downs Racetrack in Kentucky.

2 Preakness Stakes- Held at Pimlico Park Racetrack in Maryland.

3 Belmont Stakes- Held at Belmont Park Racetrack in New York.

Below are said to be the 5 most famous "Classic" horse races in the world. However, I'm sure not every one would agree.

1 Epsom Derby

2 Kentucky Derby

3  Breeders’ Cup Classic

4 Prix de l'Arc de Triomphe

5 Dubai World Cup

American Classic Races:

1. The Kentucky Derby

2. The Preakness Stakes

3. The Belmont Stakes

4. The Breeders` Cup Classic

5. The Pegasus World Cup

There are more, but these are the top five.

Thank you. Of course I know the triple crown. However, I don't know if the Triple Crown is a classic race.

Is the Triple Crown really a classic race?

Jeez...you never heard of "The Triple Crown" consisting of the Kentucky Derby, the Preakness Stakes and the Belmont Stakes?  https://en.wikipedia.org/wiki/Triple_Crown_of_Thor...

And I'm not even a huge fan!  I don't see how anyone living on planet earth with a TV and a computer could not have heard about these 3 races.  Perhaps the greatest horse I ever saw was Secretariat, winning the Triple Crown in 1973 & making all other horses look like nags.  To the best of my recollection, the jockey never took a whip to the horse.  He just loved to run.  YouTube video...

https://youtu.be/kf8GRcSeneU