I know it's not ideal but you can always type in
m.youtube.com/#/inbox and you can access your inbox, what I did was make a bookmark of that page named "YTInbox" so I can check mine easily.
and to answer the OPs question, m.youtube.com works normal on my iPhone 5 and iPad 3 running iOS 7 B6. maybe check your safari settings, wipe your cache and cookies and stuff.