Your answer indicates that you may have upgraded from 12.10 to 14.04. If this is the case, upgrading from 12.10 to 14.04 is not supported, because 12.10 was a short term release.
I have 14.04, I am running google chrome, and I can watch youtube and twitchTV without a problem. What you should do is this:
- Backup the files on your computer to an external harddrive, or to a cloud provider such as Dropbox or Google Drive.
- Download the latest Ubuntu Version
- Place the latest version on an installation medium such as a DVD or USB thumb drive.
- Restart computer, and boot up the DVD/USB installation image of ubuntu.
- Assuming this is not a dual boot setup; do a clean install of Ubuntu, wiping the entire harddrive.
After this is done; you should have a clean and fresh setup where you should not have any issues watching twitch or youtube. You should be able to do this in both Chrome or Firefox.