The issues on the North American and European PC/Mac megaservers have been resolved at this time. If you continue to experience difficulties at login, please restart your client. Thank you for your patience!
I haven't done any public dungeons in a while on the PC NA server, but I haven't had any severe lag problems within the last week doing public dungeons on the PC EU server.
Some public dungeons seem to be very popular for grinders-- either players farming the mobs and bosses for XP, or players grinding for fragments or other items (anitiquities leads?). If you're trying to do a public dungeon that has a large number of other players running around, they might be contributing to your problem, especially if a lot of them are using gear requiring additional calculations by the server.