Invite her somewhere, find out, for example, if she can dance, say that you always wanted to learn how to dance, and ask to teach, if you don't know how, offer to study together, a joint activity always brings people closer, and disposes people to get to know each other closer, only this everything should be in play mode, with lively interest ...
Ps: it doesn't have to be dancing, maybe singing, drawing, photography, in general, what would you be interested in doing yourself, find out first what she might like, it will work I answer (easy scheme)
Upd. I learned how to dance myself in half a year, everyone danced from rumba and tango to hip hop c-walka (studied together)
Try to find other interests of this girl and hit them or is she straight from anime?
And so, fight your animophobia, because if a girl is really into anime, then it will really pop up or later. You don't have to accept anime with all your heart, at least just not react negatively to anime.