This is what the DMCA safe harbor policy was meant to deal with. Hosts are not held liable for user content if they respond to a DMCA takedown notice in a timely manner. The other half of the system is that users are supposed to be able to submit a counter notice to have content immediately restored without judicial intervention. At that point the host is immune from litigation and the original party can seek remediation in the court system.

YouTube doesn't provide an implementation of the second half of the DMCA process because they have a side deal with big media to manage royalties in exchange for a system biased in their favor.