Let's not pretend that the US is unilaterally shoving copyright law onto the rest of the world. When the US made its last major overhaul of the copyright system, it was to harmonize copyright law with what already existed in Europe. In addition, the EU already signed onto the TTIP, which is a massive concession to the media industry on copyright. That wasn't due to US influence; in fact, the US backed out of the TPP.
TTIP was NOT signed into law already. It is also a completely different treaty from TTP. (TTP has nothing to do with the EU btw).
TTIP hasn't been signed into law, but it's been written by EU member states to serve EU members' interests, so it's pretty reasonable to use it as a measure of what the EU wants.
TPP is a different treaty from the TTIP, though not very - the TPP and TTIP were written in tandem to be companion agreements, and the TPP contains many provisions that were explicitly included as concessions to the interests of EU states, even though they're not parties to the TPP itself.
For most of the world, this has been true in regards to following US laws for a long time. (DMCA, copyright law)
In the end, international law is mostly a game of (economic) influence, and the EU is economically significant enough for these companies for EU law to matter.