1. MediaFire is not to blame.  There is an enhanced enforcement effort by certain organizations (Funimation).  If Funimation detects that certain titles are at certain sites, they send out a Cease and Desist letter ordering the site to delete that file.  If MediaFire complies, no more trouble.  If they don't comply, lawsuit time in a big way.  So they comply.  The solution: -you have to find other download services, at least for a while.  (By the way, even if a file were reuploaded, after the C&D letter it would probably be auto-deleted on arrival.)

2.  Torrents: Bear in mind that torrents are the most public way of downloading a file that there is.  There's even a web site in Russia that published a list of who has downloaded what files with torrents, world-wide.  And every story of legal notices being sent to end users like ourselves that I know of has involved torrents.  So especially under any enhanced enforcement efforts, like now, be certain you know the laws in your own country concerning torrents.  In Sweden, for example, their high court ruled that even running a bit-torrent tracking service is illegal.  That's why Nyaa had to reformulate how they did things all of a sudden.

At downloading services, you are only at risk to the extent that the dl service keeps records of what ip addresses accessed their site and downloaded files, and if someone gains access to those records, as recently happened in the HotFiles case in Florida.  At present, it is probably not wise to use any dl system that is headquartered or has facilities in the US.