You use POP:

If you logon to your Webmail account via a browser to compose emails, then the sent email will be stored in the Sent folder in the webmail account. You will not see this sent copy of email in Thunderbird because POP mail accounts can only access and download from the server Inbox. Pop mail accounts cannot see any other folder on server.

If you use Thunderbird to compose and send email, then it will be stored in Thunderbird Profile, mail account Sent folder. This is on your computer and not on server.

Emails in the webmail Inbox : It is normal for Pop accounts upon download to delete the copy off the server. However, if a copy is left on server - you will see it in webmail Inbox and Thunderbird Inbox.

However, you can delete emails off the server so they are not seen in the webmail account, but if they were downloaded to Thunderbird first then they will still be seen in Thunderbird.

If you wrote an email in Thunderbird and Cc'd or Bcc'd yourself: A copy of the sent email will be stored in the Thunderbird Sent folder, in addition a copy will be sent to you, so it will appear in the webmail Inbox and will also be downloaded to your Thunderbird Inbox.