Even when leaked content is still on a site, most people find it through search. If a page showing your content appears when someone searches your name, that search result is doing much of the damage.
Google runs a copyright removal process that lets rights holders ask for specific URLs to be removed from search results. It doesn't delete the content from the site, but it stops the page appearing in Google Search, which cuts off a major route to it. It works even when a site ignores takedown notices. Google also has separate tools for removing non-consensual explicit imagery.
Privly includes Google de-indexing requests alongside its DMCA takedowns, so leaked pages are pursued both at the source and in search results.