Two potential solutions (I'm sure there are others).
1.) If you want to maximize your privacy you could simply use tor "as-is" and go to a website which will show you a mobile version of a site. For example:
2.) You could also increase your privacy risk and change the Tor user agent setting itself:
Open the Tor Browser and go to the url about:config
Say Yes you'll be careful
Search for useragent
Change general.useragent.override to the value of your choice
Note: If you are the only Tor browser with a specific user agent you'll be pretty easy to identify but it all depends on what you are doing.
Additional Note: You may want to clear your cookies. If the first time you've visited a site it set a cookie regarding your screen size that may be part of what is affecting your user experience.
I just successfully did this on OSX 10.10.5 using Tor Browser 45.2.0 changing the Tor user agent string from
Mozilla/5.0 (Windows NT 6.1; rv:45.0) Gecko/20100101 Firefox/45.0
and was able to view YouTube as a mobile browser.
User Agent String Reference:
Website to see your User Agent string remotely once you change it.