You have configured Thunderbird to check for new e-mails every few minutes, but the only status update you get is from your Inbox? The other mail folders which probably contain pre-filtered e-mails (e.g. sorted by Sieve) will not be updated?
Change the Thunderbird internal behavior with the following steps:
In Thunderbird select Edit / Preferences / Advanced / General / Config Editor
- Confirm that you will be carefull.
Type mail.check_all_imap_folders_for_new into the filter line.
Double-click the setting so that the Value changes to true