Many good and serious responses in this thread. I'll offer a childish strategy, which tragically may work against a child-brained friend.

Eye for an eye. Push toxic femininity on her. Do it just like she does toxic masculinity. Create an emotional, unfalsifiable bludgeon and beat her with it. Put her in a position where she's trying to debunk your bullshit. Everything you can, throw it back in her face gender-swapped. If she starts citing sources of news and journalists, cite fucking 4chan back at her and demand she prove it has less authoritative legitimacy than whatever hack she cited.

As petty as this sounds, it's actually a simple emotional argument. Be prepared for any possible response and have an emotional counterattack ready for her emotional counterattack. Yes, this has a decent chance of ruining the relationship.