I haven't looked at the final bill but the one that was up a few weeks ago looked to me like it would only apply to service providers that knowingly facilitated sex trafficking. A service provider that removes content upon notification or otherwise finding out about the content would not be knowingly facilitating. Some might report the content to the authorities just to cover your ass.
This was directed at backpage. They are a big Pelosi supporter, and I believe big donor to EFF, sadly.