It is a misunderstanding to say that the laws are coming into conflict. As an example, if it was against the law to drive less than 50 mph and above 40 mph on a road, you are guilty of one or both for driving on the road. Law is about one question: which, if any laws, have you violated. For those with which you are guilty, what should be your punishment. This is why the law is about people finding people guilty or not guilty. Innocence is not a part of law. Unfortunately, laws that are astutely obvious get struck down because there is enough outrage.
The DMCA was written specifically to kill fair use without actually striking down fair use. In an admiring-an-evil-genus way, I'm impressed withe DMCA.