Presumably the sent emails do show in Outlook but not in your BTMail account when using browser.
If you are using "New Outlook" and sent emails do not appear in your BTMail folders when you check using a web browser it would appear that you have set Outlook up as a POP3 account which does not synchronise the folders in Outlook with those on the BT servers that you see when you use a web browser.
You will need to set up Outlook as an IMAP account.
You can not swap a POP3 account to be an IMAP account. You will need to set up your email account again.
