How does the new leader know it went to a majority? The only entity which could confirm that is the old leader.

That is correct. The solution to this is given in section 5.4.1 (election restriction), section 5.4.2 (Committing entries from previous terms) and section 8 (Client interaction) of the Raft paper.

Roughly, a newly elected leader will have all committed entries (guaranteed by the "election restriction", 5.4.2) but it does not know precisely which are committed. The new leader will commit a no-op log entry (section 8) and after it has received replies from a majority of the cluster it will know which entries have already been committed.

Ooh, thanks. The no-op commit is particularly interesting.