Well this got weird.
The first arc follows up the Batman/Superman Supergirl story when our title heroine resurfaced. With no memories and naked (yes). Which gets me right to the biggest issue in this book, not just in this first part: the depiction of women through the art. Supergirl is a teenager, it's stated more than once that she's sixteen. Yet, she is very sexualized, wears a costume leaving most of her skin bare, and she's drawn in many exploitative poses.
She's also extremely skinny, often in a way that defies how human (or Kryptonian) bodies work. There's nothing wrong with being thin but this is too much. She's drawn as visibly younger than the adults and the fact that she sexualized at the same time is disturbing and extremely inappropriate.
Other female characters don't fare much better either. Starfire is in her well, trashfire "costume" there's a lot of objectification with basically all female characters.
It's a shame because the art is actually nice, especially the colors and the inking.
Story-wise, things are better. It's nice to see Kara interact with so many other heroes (and some villains) but it does make the story very busy. I didn't like how she was in conflict with everyone, getting into fights and so many were antagonistic to her for no good reason. There's an explanation later but it's serviceable at best. Her meeting Power Girl was interesting I wish there was more of that.
Another questionable thing is the first issue refusing to give her narration in her own book, instead it's Batman and Superman who get to narrate. While observing her, despite her explicit wishes. To the writers credit this is shown in a bad light, Supergirl figures it out and calls them on it. Still their behaviour toward her is patronising. Particularly Superman's, Batman keeping tabs on other heroes is in character, especially someone just as or even more powerful than Superman.
Kara's personality was mostly fine, she's capable, a good fighter and despite her lack of experience and memories isn't easily tricked. There's a little much of adults trying to portray a teeange voice without success. Her instant crush on Nightwing was cringy to read.
Once the final conflict is resolved, it seems she found her place among the heroes and she's ready for new adventures.
This is where the problems start. The next storyline is one year into the future, in some kind of post apocalyptic world. No context is given so I don't know exactly what storyline this is, probably some crossover. Would it hurt to explain? Honestly I skimmed most of this, so idk if it was good. Seemed very convoluted.
Then there's a little dealing with the aftermath and then Supergirl is at a bar, acting like an adult while it's stated yet again she's sixteen. She's also in high school but hates it and she'd rather hang out with the 20 year old dude she met at the bar. How appropriate, nothing creepy to see here. There's also some unoriginal high school drama and bullying, ending with Kara revealing she's Supergirl and bailing on high school. The last issue is Jimmy Palmiotti/Justin Gras story with Amanda Conner's art. There's definitely way less sexualization and she's drawn like there's enough space in her body for organs. She also gets to meet Terra, who is a character I always like to see.
Take away the sexualization of a teenage hero, you get an entertaining Supergirl run with some crossover related problems. The thing is though, it's very much there. Not just in the art but she keeps saying she's only sixteen biologically but her mind is older or what. I think this is just the writers trying to get away with making an underage character sexy. It doesn't work and makes things even worse. It would've cost zero $ to make her an adult, or even better, not portraying her in a sexual manner. The costume is also questionable. Supergirl herself considers changing it at times, so maybe it was an editorial decision the creative team couldn't alter. Or they're again just trying to avoid criticism because if the hero herself thinks it's too revealing but decides she's okay with it then it's above reproach. If she was an adult then fine, but there's no get put of jail free card for sexualizing teenagers.