No, it didn't. What it did was protect online service providers from liability for either hosting or removing allegedly infringing content if they took certain action when notified of potential problems from either side (to the extent the results seem asymmetrical, it's because the pre-existing legal risk was asymmetrical.)

> SOSTA did it for illegal advertising

SESTA/FOSTA was kind of the opposite of DMCA; instead of creating a safe harbor that protected firms from existential risks that were starting to drive them out of hosting user content, it creates new holes on an existing safe harbor and drove providers out of carrying broad categories of user content around the area legally targetted.