First off the outbox is where messages go until they are sent. Then they go to the sent folder. If you set your account for POP protocol it only connects to the inbox on the email server. If you want all your folders to sync you have to use IMAP if the provider offers it.

The Outbox in TB is for storing messages to be sent at a later time. If you set up the account as POP, rather than IMAP, only the Inbox will be downloaded. If you send a message from your local laptop, copies are stored in the Sent folder, which is automatically created as soon as the first message is sent.

If you set it up as POP, set it up again as IMAP if you want to see all folders (assuming IMAP is offered by the hosting company).