Factual error: The race ends at Damascus, which is nowhere near the ocean, yet is placed on the seacoast.
Factual error: The sand dune scenery prominent in most of the movie is nothing like the real scenery along the route of the race - there are no significant sand dunes.
Factual error: Being an endurance rider myself, I know that horses, especially while racing for long distances and in the desert to boot, cannot survive with one water stop a day. Also, in real life, the horses would be very sweaty and very dirty. Because their guts require constant food horses with that little food would develop ulcers and be unable to continue.
Factual error: The actress who plays Annie Oakley is much too old. She looks (and is) in her 40's. In 1890 Annie Oakley would have been just 29, and in pictures from the time she looks like she's in her late teens. They could have done a better job with this.
Answer: They take the camel roads, which are easier (and faster) to navigate.
Xofer