I’m not a Jew, but I did live in Brooklyn for quite a while.
“Jewish law” is kind if a misnomer. Orthodox Jews follow, or try to follow, 613 commandments, mostly in Exodus and Leviticus. Disputes among religious Jews have been settled by religious courts for millenia, and to the extent that the decisions of rabbinical courts have been preserved, rabbis spend their lifetimes studying them and applying them.
Those decisions form a sort of common law, but of course there are conflicts and variances among them, with no “Supreme Court” to settle them.
So there’s written (Torah) law, applied law (rabbinic courts) and (in some countries) civil laws that apply to Jews. All that together could reasonably be called “Jewish law”.
You have forgotten the Talmud, which has governed Orthodox Judaism for at least fifteen hundred years.