phrase that transgresses the rules of grammar
Factsheet
As pointed out in @Edwin's comment referring to this earlier Answer, the relevant term here is...
Enallage
...related to solecism (a deviation from conventional word order). Enallage, however, is usually regarded as a deliberate stylistic device, whereas a solecism is commonly treated as an error of usage.
A few famous examples from advertising / TV / movies...
Got milk?
Eat fresh
We was robbed
Mistah Kurtz—he dead
Thunderbirds are go
"All of these stick in our minds because they’re just wrong—wrong enough to be right" (Mark Forsyth, Rhetorical Reasons That Slogans Stick, The New York Times, November 13, 2014)
"Solecism" is the closest thing that comes to mind. And "Malapropism" is the mistaken use of an incorrect word in place of a word with a similar sound.
Hypercorrection
It can even influence a language: Fetus is the original Latin term, but it got "fixed" to foetus or fœtus both in how people wrote Latin, and also in languages that had a term derived from it so now foetus is sometimes found in English, Dutch and German and also fötus in German; all based on people "fixing" what wasn't broken.
In between personal hypercorrection and something introducing a new spelling that became common across four different languages, are myths like "the passive voice should be avoided", "don't start a sentence with a conjunction", "don't start a sentence with however", "don't split infinitives", "don't split any verb form", "don't end a sentence with a preposition", "don't use the genitive of an noun that represents a non-living thing", "don't use like as a conjunctive". Here not only has someone got a bizarre notion into their heads about what is correct, but the "rule" has ended up being actively taught.
Often it's combined with an incorrect understanding of its own terms, as in this notorious example:
source
Not only is this teacher enforcing an imaginary rule, but of the 10 cases marked as "P.V.", only three are in the passive voice. Of those 3, only one seems like it could be rewritten into something that didn't weaken it (and that one still wouldn't actually improve). One actual use of the passive voice isn't marked as such.
In theory, such a "rule" could become so popular, that it became a real rule just by dominating the thinking of those who spoke the language. Many though are actually impossible to follow consistently, or just too at odds with common literate use, to ever reach that point.
Hypercorrection.
Wikipedia has:
In linguistics or usage, hypercorrection is a non-standard usage that results from the over-application of a perceived rule of grammar or a usage prescription. A speaker or writer who produces a hypercorrection generally believes that the form is correct through misunderstanding of these rules, often combined with a desire to seem formal or educated.
Linguistic hypercorrection occurs when a real or imagined grammatical rule is applied in an inappropriate context, so that an attempt to be "correct" leads to an incorrect result.
Hypercorrection is sometimes found among speakers of less prestigious language varieties who produce forms associated with high-prestige varieties, even in situations where speakers of those varieties would not. Some commentators call such production hyperurbanism.