The Turn of the Screw alludes to two famous Gothic novels that resonate with the governess's narrative: Ann Radcliffe's The Mysteries of Udolpho and Charlotte Brontë's Jane Eyre. The governess mentions these novels in her own narrative shortly after seeing the ghost of Peter Quint for the first time, asking herself:
Was there a "secret" at Bly—a mystery of Udolpho or an insane, an unmentionable relative kept in unsuspected confinement?
The mention of Udolpho directly refers to The Mysteries of Udolpho, which Ann Radcliffe published in 1794. In the novel, the female protagonist is taken to a large and spooky castle after becoming an orphan. Once she's there, she experiences a number of unsettling and seemingly supernatural events—the sort of events one would expect to encounter in a haunted castle.
When the governess refers to "an insane, an unmentionable relative kept in unsuspected confinement," she alludes to Jane Eyre. Specifically, the reference is to Bertha Mason, a supposedly mentally unstable woman from the West Indies whom Mr. Rochester (Jane Eyre's love interest) married years before Jane met him. Mr. Rochester keeps Bertha locked in an upper story of the house, but she finds her way out of this confinement several times and wrecks havoc on the domestic surroundings, startling Jane and putting her on edge about the secrets locked up in Rochester's house.
These allusions imbue Bly with a sense of secrecy, as the governess is left to figure out what, exactly, is going on around her. Interestingly, the allusions also arguably contribute to the possibility that the governess isn't as stable as she appears—after all, she makes some pretty sensational and farfetched conclusions after simply seeing a figure standing on one the house's towers. It's certainly strange that she hasn't seen the person before, but it's also somewhat telling that her mind goes straight to these two famous Gothic novels, which are full of an ominous kind of mystery: it's possible to argue, in other words, that she's letting her imagination run a bit wild in this moment. It's also worth noting, however, that the novella is written in a way that encourages readers to go along with the governess's logic until the end of the narrative—these allusions don't seem all that outlandish when they first appear, since readers haven't yet had any real reason to question the governess.
