Plot hole: SPOILER! Jamie Dutton is blamed for Beth being sterilized. But shouldn't the doctor be blamed for not informing the patient before the operation?
One Hundred Years Is Nothing - S5-E1
Continuity mistake: How is it that Carter has grown so much in 9 months? He looks 2 years older.
Grass on the Streets and Weeds on the Rooftops - S4-E10
Plot hole: Beth surprises Jamie in the act of disposing of his father's body at the train station. But Beth doesn't learn about the train station until season 5, episode 8. So, how did she know where to catch Jamie in the act?
Character mistake: The Native American brother points to the Bronze Star and refers to it as the Silver Star.
Continuity mistake: In the previous episode Dutton fired all the woman cowboys and kicked all women from the bunkhouse. Early in this episode Teeter is visible working as they round up cattle on the ranch. Yet a few minutes later she comes to John and Rip begging for her job back.
Suggested correction: In the scene you're referring to, the girl you see is not Teeter. It's a girl with Travis' crew.
The Apocalypse of Change - S5-E10
Other mistake: S5 E/0: The Apocalypse of Change - Tate is wearing a ball cap with a buffalo on the front, which would be fitting since his family's business is cattle ranching. Since the Dutton family are proud residents of the State of Montana, it's strange that Tate would be wearing a ball cap with the State flag of Wyoming. (00:15:08)