Can you give an example of a web site (or preferably a URL) that you're unable to access?
There are all kinds of reasons why a site might be unavailable - you haven't said whether you think the problem is limited to your browser (does it work in Chrome? or FireFox?) your machine (does it work on other machines in the same local network?) or your location (does it work with the same machine/browser in a different location?)
Here is an example of a web sitye that won't open. I clicked on it as it was a downloasd link for a song I liked posted on Youtube. OS X was blamed for the problem siting that the beginning wsytd adress couldn't be read. I was using Safari as my browser when the issue occoured.
My gosh what's the problem??
In this case you're trying to download a link using the 'wsytd' protocol. This is not a standard protocol at all, but appears to be something related to WonderShare, a third-party app for downloading content from YouTube.
As such, the problem is not with your machine, or Safari at all, it's with some kind of Wondershare app.
Either Wondershare isn't installed on your machine (so the OS doesn't know to pass wsytd:// URLs to it), or the app is broken in some way. A quick search on the net finds this discussion which indicates that you're not alone and alludes to the possibility that YouTube have moved to block this app from downloading content (presumably on some copyright basis). If that's the case then your recourse is to go to the software vendor to look for updates that might work. Either way, it's not your Mac, per se., that's at fault.