I can understand Kara's reaction, but she is a bit myopic about the whole thing here.
And so is Lena, because she doesn't realize that Supergirl has emotionally valid and understandable reasons for her reaction. She's not just being imperious or having a god complex. The whole thing that makes this debate so intriguing is that there is no simple "This side is right and the other is wrong" conclusion. There are equally valid arguments on both sides. And both Supergirl and Lena could stand to try to understand each other's perspectives more and recognize why they think the way they do, rather than jumping to negative conclusions about the others' motives.
I'm starting to wonder if this is building up to Kara revealing her identity to Lena as a way of regaining her trust. Maybe Lena would be more willing to hear Supergirl's concerns if she knew they were coming from a dear friend whom she trusted. Although that could easily backfire and make the rift worse.