Why wouldn’t she be?
Kara seems to be susceptible to Bizarro’s powers across the board. It’s not only Bizarro’s flame breath that can be presumed to be harmful to Kara, but her freeze vision as well.
Kara expends a great deal of energy blocking Bizarro’s eye beams, which she wouldn’t do unless she thought she could be harmed by them. In fairness, she probably would have assumed at that point that it was ordinary heat vision, but the fact that she blocked them indicates that her own heat vision might harm her (at sufficient strength). If this is the case, the ice vision (being equal and opposite) is presumably equally powerful.
In the comics, we see something similar:
Supergirl blocks Bizarro’s flame breath and dodges her eye beams. Clearly she thinks they might be dangerous—which wouldn’t be the case if she were completely fireproof.
At a more basic level, this is not really surprising. If an ordinary person walks up to Supergirl and punches her, she’s barely going to notice it. On the other hand, if Bizarro punches Supergirl, she might actually injure the latter:
Supergirl is not punch-proof: she’s highly punch-resistant. She’s invulnerable to most physical attacks because the energy output of those attacks is so small relative to her toughness.
So there’s no reason to expect her to be fireproof: she’s fire-resistant. Sure, an ordinary candle flame isn’t going to hurt her, and nor is a five-alarm fire, as amply demonstrated earlier in the series. Maybe even flying into the sun won’t. But Bizarro is also super: her flame breath can be much hotter than anything Supergirl has experienced before: hot enough to burn Supergirl, just as Bizarro’s punch is strong enough to injure her.
Indeed, according to the (admittedly unreliable) DC wiki’s description of a different Bizarro,
Bizarro’s flame breath may actually be “super-heated plasma or lava-like fluid.” The former in particular can get a lot hotter than most house fires.
The energy output of Bizarro’s powers and that of a mundane fire are vastly disparate. Thus one can be no more than a nuisance while the other poses a real threat.