I don't think she should close fan sites down, even if it was possible to do so. It would just inspire more hate.
What I think she should do instead is freeze them out. Give exclusive content to any fan site brave enough to stand with her, or do giveaways. It would be really lovely if she sent signed books to fans who write to her in support or something.
Jo Rowling isn't a petty person. She's not mean-spirited and I hate to see this becoming the public perception of her. The only way to fight it is to rise above it, imo.
The Ickabog is one of the loveliest, funniest books for younger children I've read in a while. A little bit Road Dahl, a little bit fairy tale, a little bit Animal Farm. I hope it does well when it's eventually released, especially with the proceeds set to go to those affected by coronavirus.
It's been wonderful to see Jo interacting with all the kids who drew illustrations for the book on Twitter. She uses each of their names and invariably picks out one particular thing to praise about the picture, to give her response the personal touch. I can't imagine how thrilled the kids must be to hear from her. It warms my heart, it really does.