Though Twitter rolled out its redesigned version for web and mobile versions in Dec 2011, still web version not enabled for lot of users, yesterday my twitter handle got the new Twitter, since then I‘ve been looking for ways to move the twitter timeline/content pane on right to left side of the screen, finally found a user script that does it on Chrome and Firefox.
Fix Twitter: Move Content Pane Left User Script
All you need to do is- click on this link and allow it to install in Chrome, though you can install this script in Firefox with Greasemonkey installed, but its not working in Mozilla browser.
Tested in Chrome, its working. Now I can see Twitter timeline as usual on left side of the screen. If you want to access New Twitter again normally that Twitter offers simply disable the script we installed in Chrome extensions page.