When portal users open a website page, the backend navigation bar doesn't show even if the menu data is correctly send to qweb context. It's a security issue, due to the groups attribute of the user_navbar template. The share module override the webclient_bootstrap template definition by adding the portal group and the share group in the template attributes, but website use another template (user_navbar). I think this is because website and portal are two independents modules and shouldn't assume that portal group or website template exists.

The solution is : Settings => User Interface ==> Views : search for user_navbar and add "Portal" in the groups tab.