Other mistake: During the boardroom meeting with Lena, James, and Morgan Edge, Katie McGrath's natural Irish accent briefly, but noticeably, slips out.
Other mistake: During an argument, Maggie's father mentions the border wall being built to keep people from entering the country illegally. This is obviously a reference to the real-life presidency of Donald Trump, but within the show's continuity, his comment doesn't make sense: in "Resist", the penultimate episode of season two, it's revealed that President Olivia Marsdin is literally an alien from outer space, and earlier in the season in "Welcome to Earth", she signed an executive order offering aliens such as herself full citizenship if they so desired it. If she's that tolerant of extraterrestrials, it seems unlikely that she'd be so intolerant of foreign immigrants as to resort to such an extreme action in foreign policy like building a wall on the border.
Plot hole: After Kara and Alex are run off the road, Kara is convinced by "Noel Neill", revealed later to be J'onn, to stop using her powers. From context, J'onn had to have been summoned by Eliza, but in 2007, she would still think he's just Hank Henshaw; she doesn't learn he's not, or even that he's a shapeshifter, until 2016, as seen in season 1's "Myriad", well after Kara and Alex have grown up.
Continuity mistake: As Kenny is about to sit down opposite to Kara, the juice on his tray is lying down on it. Next shot it is standing up.
In Search of Lost Time - S3-E15
Continuity mistake: When Samantha sees the surveillance video of her transformation into Reign, which was triggered by Lena provoking her, their dialogue is slightly, but very distinctly, different than it was a few minutes earlier. (00:20:15 - 00:25:25)
In Search of Lost Time - S3-E15
Revealing mistake: During the mental block Alex stops the fight between Winn and the Asian guard then she hits him twice. The punch to his face misses by inches but he falls anyway.
Shelter from the Storm - S3-E18
Continuity mistake: When Winn and Brainy hug, Winn starts the hug with one arm high and one lower, towards Brainy's waist. The shot cuts and both his arms are up and in the centre of his body.
Not Kansas - S3-E21
Revealing mistake: As the guy fires the machine gun at Guardian, there's a closeup of bullet casings falling to the ground, but they're not just empty casings, they're fired blank rounds.
Continuity mistake: The shield design that Winn sketches on the paper changes between him starting to draw, and him showing it to his friend a second later.
Continuity mistake: The sketch that Brainy shows to Winn is meant to be his original sketch that he drew in the previous episode, hence why Winn is so shocked he's somehow got it. But it's slightly different from what we actually saw him draw at the time.
Answer: For one, it's usually cheaper to film in Vancouver because of the tax breaks. But all the CW Arrowverse shows (with the exception of "Black Lighting") are/were filmed in Vancouver. So it makes it easier and cheaper for sharing set designs, crossovers, etc ("Black Lightning" filmed in Vancouver during its crossover). Originally "Supergirl" wasn't part of the Arrowverse when it was on CBS, but it was too expensive to produce and wasn't reaching the wider audience the CW shows had. So CBS moved it to its sister company, The CW.
Bishop73