Normally this only happens with shared mailboxes connected to your Outlook.
To resolve this issue, try adding the following registry key:
HKEY_CURRENT_USER\Software\Microsoft\Office\x.0\Outlook\Preferences
(x.0 is the Office version installed on your PC, this can be 16.0, 15.0….)
Add a new DWORD called DelegateSentItemsStyle and give it value 1
After this, restart your Outlook and the problem should be solved when sending a new mail with your shared mailbox.
But if you now have the problem where sent items get stuck in your Outbox, but still they go through to the recepient, you will have to preferably enable cached mode in Outlook or delete the DelegateSentItemsStyle registry value (or change the value to 0).
Sources:
https://support.microsoft.com/en-us/help/2843677/messages-sent-from-a-shared-mailbox-aren-t-saved-to-the-sent-items-fol
https://support.microsoft.com/en-us/help/2703723/email-remains-in-the-outbox-when-you-use-the-delegatesentitemsstyle-re