Recommendation letter process
Having the letter of recommendation (LoR) writer send the LoR directly to the institution to which you are applying is indeed the standard procedure in the English-speaking world and increasingly beyond. In this procedure, the LoR is confidential; only the issuer and the recipient(s) should normally see it.

The process works as follows:


Online forms usually have two "sides". The first side is for you, the
applicant. You enter the name and email address of your LoR writer,
after you have obtained their approval. An email will be sent to the
LoR writer with a link to "their" side of the website, to which they
can then upload the letter and perhaps additional information.
For the offline process, you write your application, and the LoR writer sends in their LoR independently from you before the deadline. 


However, in some countries (until recently e.g. in Germany and Austria) a different procedure is used, in which the LoR is handed to the applicant who then submits it together with their application (via website or offline).

I would contact the HR department at the institution where you are applying and ask for clarification, then talk to your LoR writer again. If the institution explicitly requests that you hand in the letter, you can just relay this information to your supervisor. You may also want to make your supervisor aware of different traditions of LoR writing in different countries. Alternatively, ask the HR department whether they can accept a LoR directly from your supervisor.