Yes, although I thought that was a bit of a plot hole in the last episode; Livewire (before she becomes Livewire) was ranting on her radio show about Supergirl and called her “adorkable”. The adorkability is really a Kara thing.
That’s why I called it a challenge; in order to craft a plot worthy of a genius supervillain, you need, at least, very smart writers. Supergirl’s writers may not be up to that challenge, yet. The fact that villains have gone against her strength can be handwaved a bit; Supergirl is new on the scene and the baddies don’t really know how best to take her on. On the other hand, Superman is known and they might well assume that sauce for the gander is sauce for the goose.
Does that apply to the secret identity issue as well? If it’s known that Superman has an alter ego (even if it’s not known that he’s Clark Kent), the folks in-universe might assume that Supergirl does the same.