It was at this time that women were just starting to ride astride again after centuries of riding sidesaddle. I'm not sure Elizabeth would have actually been riding astride yet, but anyway, here's an interesting article on the subject:
Sidesaddles and Suffragettes
The Fight to Ride and Vote
by CuChullaine O'Reilly FRGS
http://www.lrgaf.org/articles/sidesaddles_and_suffragettes.htm
I feel like she was shown being uncomfortable on horses early on in the show, so it seems believable to me. I assume she didn't have much or any experience with horses previously, so, paired with her tendency to try to be tough like the guys "I can it just as well as you", I could see her riding astride vs side saddle.
I don't know how much women were riding side saddle out in the "west" (US or Canada) and rural country. It's not at all practical.