From the Opera Website (bold is mine):
"The technology behind Opera Turbo is a proxy server with server-side
compression of webpages. A compression rate of up to 80% can be
achieved, in part by reducing the quality of images.
My guess is that your provider is having issues reaching some of the content servers you are pointing to. The "compressing proxy server" your browser is pointing to when you are using "Opera Turbo" is not among those. So, your provider - faulty to get somewhere on the net - reaches successfully the Opera Turbo server, which in turn gets all the pages from wherever you want and gives them back to you; so you're able to reach all of the net from that source.
To verify if this guess is correct I would try to use the Tor browser: https://www.torproject.org/projects/torbrowser.html.en
If that can leave the Internet and get you everywhere, too (although at a way slower rate than with Opera) that would confirm this. The only thing you could do then would be give a call to your ISP.
Also, theorethically you could have some kind of firewall that is preventing you to leave the Internet towards certain sites (not the Opera Turbo proxy, see above), but that is less likely.