Not true. If you're alerted to offending material and you remove it, you haven't broken safe harbor status what-so-ever. Otherwise YouTube, DropBox, Instagram and countless other sites would be dead.

The only way you break safe harbor is if you know of offending material, eg by a filter or having been alerted to it, and leave that material in place. If you run content filters (such as at the point of upload), then you have to be more careful.

The difference being between after-the-fact moderation and pre-screening. If you pre-screen you take on a greater burden of risk, in terms of copyright holders being able to argue you knew about something and didn't remove it, or should have known.