The lady doth protest too much, methinks" is a quotation from the 1602 play Hamlet by William Shakespeare. It has been used as a figure of speech, in various phrasings, to indicate that a person's overly frequent or vehement attempts to convince others of something have ironically helped to convince others that the opposite is true, by making the person look insincere and defensive.
In rhetorical terms, the phrase can be thought of as indicating an unintentional apophasis—where the speaker who "protests too much" in favor of some assertion puts into others' minds the idea that the assertion is false, something that they may not have considered before - Wikipedia