Continuity mistake: It is the evidentiary hearing, all characters are put on the witness stand so you get to see what all of them are wearing. The trial starts the next day and lasts at least three days, but when they are in court at the end of the trial for the reading of the verdict for the gravedigger they all have the exact same outfits on as the day of the evidentiary hearing.
Other mistake: Episodes 4 and 5 of season 1 aired noticeably out of order. In episode 4, "The Nanny," Alex's former Nanny, Rosalba, comes to work at his podcast company, and its name, "Ajana," is mentioned a few times. But episode 5, "The Mother-in-Law," starts with the company unnamed; over the course of the episode, Alex learns of the concept of "ajana," the unknown, from his mother-in-law and chooses it as his company's name.
Continuity mistake: When the two factions of Hondo's pirates are fighting each other, the ones who sided with Maul and Savage are using blasters that fire red bolts, while the ones that stuck with Hondo shoot blue bolts. When Hondo persuades the traitors to work for him again and they attack the two Sith, all of the pirates' blasters are now firing blue bolts. This is Star Wars, not G.I. Joe, and weapons don't just change colour like that.
Factual error: While discussing the death of a car accident victim with detectives, the pathologist Dr Elizabeth Rodgers takes out a french fry, seasons it with mayonnaise and eats it. She is in the morgue, wearing bloodstained scrubs. Nobody ever eats or drinks in a morgue. This is not a character error; eating and drinking in a sterile laboratory environment is absolutely forbidden and this is taught to medical and science students from day one of their degree courses. In fact she would not even have food in there in the first place.
Revealing mistake: Watch the opening sequence of this episode very carefully - Clarkson, May and Hammond are not in Downing Street (they are on a set that stands on the BBC backlot) and library footage of the Prime Minister is cleverly edited in to make it look as if they were there at the same time. Even taking into account the fact that the Prime Minister's security service would never allow such an intrusion onto Downing Street, the line up of the shots does not quite match - our three intrepid heroes should be visible in shot when the Prime Minister is shown exiting number 10, but they are nowhere to be seen. It is all sleight of hand and clever editing, but detectable if you watch carefully enough.
Other mistake: As Young Sheldon is a prequel series to Big Bang Theory, this episode is at odds with episode 2-13 of Big Bang Theory, "The Friendship Algorithm," in which Sheldon is unfamiliar with books like "How to Make Friends and Influence People," which he reads in this episode of Young Sheldon, produced nine years later but taking place twenty years earlier.
Continuity mistake: Up until Green No More in Series 2 (where the Green Ranger powers are destroyed), the Green Ranger's overshield constantly changes between solid-looking metal to flimsy-looking cotton. This is the most obvious difference between the Japanese and US footage.