I don't think that's true at all.
Civil libertarians and the legal sector have a massive voice in the Democratic party; the establishment still runs the show, not the progressives. There's a reason why even in deep blue states like California, New York, Illinois, or Massachusetts, there's no momentum in their state legislatures to pass any law cracking down on "internet misogyny" or "online hate speech".
Even if such a law garnered the necessary votes to pass, whether at the state or federal level, the courts would immediately strike it down. Members of the federal judiciary, liberal or conservative, have consistently ruled that 1) content-based restrictions on speech are subject to strict scrutiny, and that 2) there is no "hate speech" exception to the First Amendment.