You can certainly minimize your problems but you will not "fade in history". With sites like http://archive.org/web/web.php data about your website will usually be available for someone who wants to find them. That being said, a company is much more likely to spend their efforts trying to stop a current infringer than go after a site that has since stopped.
Bring it down or move to servers off shore or where they do not care http://www.cyberbunker.com/web/index.php
1. Prepay for your hosting as far out as possible; 2. Put all your money in a safe deposit box at the bank; 3. Change your name; 4. When they come knocking move to a country with no extradiction, maybe Ecuador.
It depends on your luck, if the copyright owners are not tracking you at the moment because you aren't that popular, you MAY be able to get away with it. But that is a big MAY. Unlike civil cases where there can be a limit to the number of years for prosecution, criminal acts do not expire.
In US law there are statutes of limitations for criminal prosecution. For example, many states have similar statutes to these: murder charges: no time limit serious felony charges: six years misdemeanor charges: two years, and petty misdemeanors and infractions: six months. There are things that can toll the statute of limitations, but many criminal acts do end up beyond prosecution because of the passage of time.
Illegal is a matter of location. There must be some countries where this activity is not illegal and maybe even protected. Just a matter of locating your hosting there. Maybe cloud hosting. The next problem would be protecting yourself from being identified and prosecuted depending on what country you are a citizen of. Do some sort of risk assessment. Getting "busted" for illegal activities is just part of the risks. The gains might be enough to justify the risks. Your past history might also be a factor. If you are currently running an illegal site and just now wondering what to do to protect yourself you might have things backwards. That planning should have been done first. You might also determine what are the real/actual penalties if you are "busted". That's all part of the risk assessment. How many people doing this sort of thing have been caught? It is just a business with the added factor of possibly being illegal.
My inbox has space, if you need my advice only. Maybe I know few people can help you with that giant site of yours.
Most illegal website operations are hosted offshore and the owners cover up their tracks well to avoid being prosecuted. If you've been careful, you should be fine.
You were not careful and are not fine, you were caught. (banned) Hopefully this does not happen to the OP.
However much money you make... I believe the copyright people will fine you much more. Because the piracy business model makes money because people can buy materials from you cheaper right, eg a 10th of the cost. So for every thing you sell, you may get fined 10 times as much as your sale or profit if the copyright people find you. I am just roughly estimating here. I think this is a bad business to be in personally. Even if I could think of things that would prevent you from getting caught I probably wouldn't tell you, for ethical reasons too. As well as the fact that a copyright troll or agency might come after me for helping you... probably not likely but why would I take the chance.
I would advice you to stop whatever you're doing, you might be lucky now. The law always catch up with those illegal operations when they get famous. Unless, you intend to play "hide and seek" by hosting offshore and taking extra precautions etc..
This is now my new favorite question of all time. OF ALL TIME! Here is the best course of action to protect yourself from legal liability and consequences: Find and hire a lawyer immediately.
TorrentFreak (torrentfreak.com) does a great job of providing the latest in "scene" news and describes the consequences of such illicit actions.
first off, if it's yours, why are you hiding it? "Uhh yeah, it's uhhh... a friend of a friend's"?.? Secondly, the design of your site screams illegal. Thirdly, why are you using trademarked logos such as IMDB? Forthly, are those images public domain? You're screwing yourself every which way to sunday. If you're going to link to "knowingly" infringed material, then just do that (or don't at all). Don't make your shit look so dark.