In Britain, “normal” English is well defined as being the accent taught to the upper class. In America there are regional accidents, though mid-western (mid-Amerca) became standard as a result of the civil war (when someone from Illinois accepted the surrender from a Virginian). Of course, America has been very open to absorbing words from a variety of languages (and to a lesser extent, British English as well), including Yiddish. One can argue that whatever people are using is by definition “normal”. So normal English includes that spoken in Brooklyn.
Note that some countries (France and Israel come to mind) have government agencies defining what is “normal”, though it should be noted that real people tend to speak without regard to bureaucratic requirements as to what is normal. To most Americans, the idea of an official “normal” English is laughable.