Most web pages are not designed for 1080p screens, as not everyone has one yet. There is most likely CSS limiting how wide the text can go, so that people with smaller screens (1024x768) can still read the web page without having to scroll left and right constantly.
Using a tool like Firebug (firefox) or Chrome Developer Tools you can manually edit the css, but you'll need to do this for every site. I doubt an extension/plugin exists to do this automatically, since it would be difficult to detect what div (a css thing) needs to be widened. Difficult, but not necessarily impossible.
In any case, my personal recommendation is that you either zoom in a bit (Ctrl - +), or do 2 things at once, and have your browser on the left side of the screen, and some other thing (another browser?) on the right side. Otherwise, there's not a whole lot you can do.
For the record, zooming does not really make things all that blurry, unless the images are VERY low quality to begin with. The text scales pretty nicely, so I'd say that is the more preferred option.