If you’re using IMAP, the Sent folders in Thunderbird and the web applications aren’t the same.
So if you delete it in Thunderbird it won’t necessarily be deleted in the web applications.
Webmail and Thunderbird can perfectly use the same folders for “Sent” and “Trash”.
In each of the two programs, make sure to “subscribe” to all folders to see them all, then decide how to configure the identical folder from both platforms.
Note that if Outlook was involved, it will certainly have created a whole slew of folders in French on the server, and that is a nuisance.
I agree with what @fritz2cat said; it has happened to me as well that sometimes Thunderbird hasn’t subscribed to all folders, confused by the switch from Outlook.
To check whether you’ve subscribed to all folders, right‑click the account in the left‑hand list and then select the “Subscribe…” (Sottoscrivi…) entry. Once the subscription window opens, click the “Refresh” (Aggiorna) button and make sure all folders are checked.
If that still doesn’t work, go to “Account Settings” (Impostazioni account) in Thunderbird’s main menu and, under the “Folders & Copies” (Cartelle e copie) section of your account, ensure that the folder associated with Sent Mail is the correct one. If you have more than one, try a few alternatives.
Side note: The “Subscribe…” (Sottoscrivi…) entry does not appear if you right‑click while the “Unified Folders” (Cartelle unificate) view is active. If that’s the case, switch to the classic view before proceeding.