SharePoint has always had a bit of a “Design Issue”. Let’s be honest. SharePoint has never been accused of being the hippest kid in the classroom. But the built-in design for displaying content on mobile devices is downright miserable. As a consequence I always recommend my clients to disable that and go for a responsive design or a specific mobile design. This, however is not an option, if you are on SharePoint Online, via Office 365. Well there is a workaround for achieving the disabling. Check out this script for more intel. If you are on the SharePoint Online 2013, you should be able to just use the device channels to control your design.
Here is a great post on how to disable mobile redirect for SharePoint 2010 sites: Shutting Down the Mobile Redirect for SharePoint 2010 Websites | TCSC Blog.