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In Kara’s defence, there was a lot going through her brain at the time. Firstly she was feeling guilty about the whole secret identity thing, especially since from Lena’s perspective she had a friend she was getting along with and a superhero who seemed tense and guarded around her. Kara didn’t like the way she acted as Supergirl to Lena, but she knew if she didn’t censor herself she’d end up saying or doing something to expose her identity. She wanted to tell Lena, but whenever she brought it up J’onn would get this disapproving look on his face and Alex would go on another you-can’t-trust-her rant, besides, she wanted to tell Lena when she was in a good place with both of her identities first. Secondly, she was stressed about Reign and the other Worldkillers and how to stop them. Thirdly, she was panicking after just finding out Lena could make kryptonite, and fourthly, she was in a complete gay panic surrounding her decidedly not platonic feelings for her best friend. So yeah, a lot going on, so can she really be blamed here?
Everything happened just after she found out about point three, and her and Lena began arguing shortly thereafter.
“You need a way to stop the Worldkillers, and I’ve proposed one, why are you so against it? Is it because I’m a Luthor and you think I have ulterior motives, or has being a superhero given you such a God complex that anything that could possibly harm you isn’t an option?”
“What? I don’t have a God complex, and you’re twisting my words. Why is it so hard for you to understand why I don’t like the idea of using kryptonite?”
Winn, who left shortly after the two started arguing reappeared next to Alex with a bag of popcorn in his hands just in time to hear them escalate into shouting. He offered the bag to Alex, who glared at him. No skin off his nose, more for him. He tuned back into the shouting match, with a handful of popcorn crammed into his mouth. Winn and Alex were far from the only witnesses to the argument, most of the agents had migrated closer to watch the show. Most were waiting for Lena to slip up and say something xenophobic or accidentally let loose a vendetta against Supergirl, no one was prepared for how the argument would end.
“You know what Supergirl, you can go fuck yourself!”
“Why don’t you fuck me yourself, coward!”
Lena paused in shock before responding.
“Maybe I will!”
“Good!”
“Great!”
There was another pause, before Kara spoke again, this time not shouting.
“Your place?”
“Let’s go.”
Supergirl picked Lena up and they vanished in a flash.
Dead silence echoed throughout the DEO, eventually Winn broke the silence.
“Did… Did that really just happen, or am I dreaming?” Alex backhanded his shoulder, “Oww, ok definitely not dreaming.”
I mean, can you blame her?