I'm using Thunderbird for my work email account, which is powered on Microsoft Exchange I think. I'm OK with sending and receiving messages. However all the sent messages are in Thunderbird only. Is there a way to sync them into my real sent box, so that I'll be able to view them on the web mail on the browser?
In short, no. POP3 and SMTP were designed back when storage was much more expensive, and nothing was typically left server-side. If your email is indeed coming from Exchange, you should be able to use IMAP, as long as your employer is allowing access to it. If your employer is running Exchange 2007 or 2010, there is also an EWS (Exchange Web Services) plugin called ExQuilla that will allow for a somewhat more native connection (I believe this is similar to how Android and iOS devices connect). If I'm understanding it correctly, ExQuilla would allow Thunderbird to function more like Outlook, including Global Address Book support.
If you do set up an EWS or IMAP connection, it should then be possible to copy/move everything from your local sent box to the server.