For the Girl Who Has Everything - S1-E13
Other mistake: In Kara's hallucinations of Krypton, the planet is shown with a yellow sun. Rao, the supergiant around which Krypton orbited, was red.
Supergirl (2015)
1 mistake in For the Girl Who Has Everything
Starring: Calista Flockhart, Melissa Benoist, Mehcad Brooks, David Harewood
More quotes from Supergirl
Trivia: The episode "Exodus" introduces Mon-El's father Lar Gand. In DC comics, Lar Gand was the birth name of Mon-El.
Question: Why did they change her costume and hair style?
Answer: I can't say for certain, so I'm surmising that the change was to "catch up" with the other Arrowverse shows, particularly Arrow and The Flash, whose lead characters have undergone various upgrades and redesigns of their superhero suits over the years; after four seasons, Kara's probably just felt a little stale. And I believe the change in hairstyle was Melissa Benoist's personal decision and had nothing to do with the character.
Answer: Another part of why the change was that it's cold in Canada in the winter and the old costume didn't protect her from it, whereas the new one is nice and warm.