The way some people see Supergirl is the perfect demonstration that feminists have been right all along: a woman does have to work twice as hard just to earn half the pay and a tenth the respect of a man.
The double standard is astounding; the same people who can watch Superman reverse time by reversing Earth's rotation without so much as blinking, are the same ones who complain that they never explained where Supergirl got her outfit. They see nothing wrong with a depowered Superman who walks all the way to the North Pole without food, water or so much as a scarf, where he recovers his powers without explanation. Then they turn around and complain that Supergirl wasn't wet when she came out of the lake. When the depowered Superman discovers the world has been taken over by a superpowered madman, he walks to the North Pole. He doesn't run, or jog, or ride a dogsled or anything; he walks there. When Supergirl arrives to Earth, she decides that having a base of operations, sleeping on a comfortable bed, and eating real food are better than sleeping in the woods eating crickets. Guess who is accused of not showing enough desperation in the face of adversity and who isn't.
Most of the flaws that people see in Supergirl exist only in their imagination. For instance, Supergirl's costume. The movie clearly shows that Supergirl can change the shape of her outfit at will. She does it when you first see her in her Supergirl outfit, then when she changes to Linda, then when she flies out the window to confront the invisible monster, etc., etc. So why is it that people complain about the wrong thing (the first appearance of her Supergirl outfit) rather than the right one (her unexplained ability to change her clothes)? Because they were not interested in watching the movie, they were only interested in finding flaws in it. But how can she change her clothes anyway? Let's see, she's a superpowered being who can move at nearly the speed of light, who comes from a civilization several hundred years ahead of ours and who wasn't exactly wearing clothes bought at Kmart to begin with. Maybe she uses her speed. Maybe her clothes can chance their shape. Who knows? By the way, is there anyone over six years old who believes that Peter Parker could've made such a stunning costume by himself when it took a small army of designers and seamstresses to make the ones used in the movie? Of course not. Is there anyone who cares? Of course not, it's just a freaking costume. Oddly enough, this reasoning, though perfectly valid when applied to Spidey, is never used with Supergirl.
Another complaint is that the movie never explains how Supergirl knows so much about Superman. Er, did anyone miss the fact that half the movie is spent with Selena chasing Supergirl all over the universe using a device from Argo City? Do I need to remind anyone that Kryptonians aren't exactly Amish?
Some people claim the movie has bad dialogues. Like when Supergirl asks a thug "why are you doing this?" and he responds "that's just the way we are." Let's assume for a moment the apparently preposterous notion that what characters say should be the result of their intelligence, knowledge, independent thought process, plus the context of the dialogue. So here's the scene: two thugs confront Supergirl with the intention of "having fun" with her. She easily throws one of them away and asks him why he is doing this (this is the first time she's ever met a "bad guy"). And what exactly do people expect his response to be? Shakespeare? A 2-hour dissertation on the psychological origins of his motivations? A one-liner so full of charm and wit that would make Jon Stewart insane with envy? This guy is neither an English professor nor a rocket scientist and the only things in his mind were the curvier parts of Supergirl's anatomy. He's not interested in impressing anyone or sounding witty. He simply says the first thing that comes to his mind, and what he says is pretty much within the ballpark of what you'd expect from him given the circumstances. The problem is that people are too used to characters in other superhero movies who are nothing more than a mouth and an attitude, plus a collection of puns and one-liners waiting to happen, where the idea of characterization is to have a guy yell "kneel before Zod!" every ten minutes.
But the silliest complaint is that she joins a school instead of going bananas trying to find the Omegahedron. Let's see, when you're looking for a needle in a haystack, what would be more productive: to be smart and methodical or to run around like a headless chicken? Supergirl chooses the first option. She finds herself a comfortable base of operations (where she can easily find all the information she needs about Earth), where she hides during the day, using the night as a cover while she searches. But why would she hide? Because when you're searching for a dangerous weapon, the last thing you'd want to do is let its current owners know you're looking for them. They'll either hide or sneak up on you. Once again, people are so used to stupid superhero behavior (e.g., Superman struggling to pull the Kryptonite necklace over his head instead of just bending over) that they're unable to recognize the one time when someone uses her brain.
In the end, there's nothing that Supergirl does wrong that Superman doesn't do ten times worse, yet all the disastrous flaws in Superman are either ignored or dismissed as nice little quirks that give the movie a lovely homemade feeling while every tiny detail in Supergirl is regarded as a crime against humanity. Is that sexism? You bet.