I'm not a wild west expert by any stretch of the imagination, but I do know quite a bit about one of them. I've read / seen just about everything there is to see on Billy The Kid and The Lincoln County War. So I'll speak about that episode.
I watched that episode first because of the fact that this is a Fox News produced show. I found what I usually found with Fox, they don't out right lie or anything, but they don't tell the entire story at all. They leave out things and take "artistic" license with a lot of it. Which is fine, unless you are running a series which is supposed to tell the "truth" rather than Legend.
Dermot Mulroney was a great choice for the narration, and most of the experts are very well respected. I don't know what makes Bill O'Reilly an expert at all, other than the fact that he produced the show.
The Billy the Kid episode was deeply flawed, for example, when they show the part of Billy's capture at Stinking Springs they make it look like a town, when in fact it was barely a building. Also they make it look like there was a big gun fight between Billy and Pat, when in fact Billy gave up after Pat and his crew opened up on Charlie Bowdre, mistaking him for the Kid. They also make a lot of assumptions about Pat knowing Billy. When in fact he barely knew him, but that doesn't make for good TV does it.
I could go on, but basically this is like everything Fox News does and should be taken with a grain of salt. There are plenty of real documentaries out there if people are interested in the subject. I'd suggest watching those for a true historical account.