Rumor has it that both Supergirl and Bizarro might be introduced in the direct sequel to Zack Snyder's Man Of Steel, and that they'll both factor into Brainiac's plan to get his hands on the codex that Zod was after in the first movie! Heroic Hollywood reports that the alien android will actually attempt to retrieve said codex from Kara Zoe-El, who did feature in the Man Of Steel prequel comic and is is generally assumed to be the occupant of that empty pod Kal-El (Henry Cavill) takes notice of when he discovers the Kryptonian ship in the first film.
Now this doesn't really make a lot of sense given there was only one codex and Jor-El (Russel Crowe) hid it in his son, but they'll most likely come up with some way of explaining that... if this info. is accurate, that is. Anyway, the site also reckons it's Supergirl's involvement that piqued George Miller's interest in the first place, and that - like Furiosa in Mad Max: Fury Road - she's expected to feature very prominently in the story.
So where does Bizarro come in? Well, apparently Brainiac uses what Kryptonian knowledge he does have - an presumably some of Superman's DNA - to create the Man Of Steel's twisted mirror image, and send him after Supergirl.
Now keep in mind there is no script for this movie yet (as far as we know, anyway) so all of these details are purported to come from the pitch Snyder and writer David Goyer gave to Warner Bros. when they were tossing around ideas for a Man Of Steel sequel. There's every chance at least some of what is detailed above will make its way into the final film, but as usual, it's all subject to change. If this was to be the premise though, would you be happy with it?