Depends a lot on how the law is worded. It's fine that content providers are obliged to take down offending material when they are made aware of it. It's quite another thing to make content providers responsible for finding offending content, and levying harsh fines if they miss something. This is what was proposed earlier in the EU, and it can lead to censorship-by-proxy if it is coupled with vague boundaries: hosting companies will err on the side of caution. Once those laws cover not only kiddie porn and harassment, but also "fake news", you're getting into muddled waters.
Also, how fast are providers expected to act once notified, and is that urgency related to the nature of the offending material? With kiddie porn you'd expect providers to act fast, but currently we see them taking DMCA takedown notices with a similar urgency, which is silly.