There’s one in every family.
Kara Zor-El was the daughter of Jor-El’s wealthy-yet-disreputable brother, making her the cousin of the infant Kal-El. On Krypton, Kara was doted on and spoiled, and she loved nothing better than sneaking into places where she wasn’t supposed to go.
That’s why she was in Jor-El’s laboratory on Krypton’s last day, watching her father and his brother, the eminent scientist and the sleazy politician, having their last fraternal argument. To avoid being caught, she hid herself in a prototype ship, a ship very much like the one she watched her little cousin being placed inside. And as the tremors began, Kara fell against the control panel activating the prototype ship’s launch sequence.
Due to the mysterious tricks of space travel, relativity and stasis, Kara’s ship arrived on earth 25 years behind Kal-El’s ship.
If I were writing a Supergirl miniseries it would begin with Clark Kent being about 25, a young superhero just starting his career, when suddenly he finds himself responsible for a very spoiled, very powerful 9-year-old.
Since she doesn’t arrive on earth as an infant, Kara would start out being much more of an alien, and having a harder time adjusting to life on Earth. And due to the mysterious tricks of Kryptonian biology, her powers develop much more quickly, but to a lesser extent than Superman’s.
The series would cover 10 years, up to the point when Kara is 19 and Clark is 35. It would end with the arrival of Brainiac, which threatens the whole Earth, and is the impetus for Kara to finally learn the self-sacrifice that marks a true hero.
Finally here's a very different take on Supergirl from a few year ago (2007, I think):