I haven't found a way to delete all chats in bulk like you can on Chatgpt. With Claude, you have to scroll to the bottom, select everything, and delete. The problem is, if you have a lot of chats, it becomes impossible. I created this script. It does it alone. I hope it helps someone.
(conversations disappear from the UI slowly, over several minutes, and remember to keep the tab open until the console shows "Finished", refreshing away from the page can stop the deletion process.)