Make the current Git branch a master branch

Make sure everything is pushed up to your remote repository (GitHub):

git checkout main

Overwrite "main" with "better_branch":

git reset --hard better_branch

Force the push to your remote repository:

git push -f origin main

The problem with the other two answers is that the new master doesn't have the old master as an ancestor, so when you push it, everyone else will get messed up. This is what you want to do:

git checkout better_branch
git merge --strategy=ours master    # keep the content of this branch, but record a merge
git checkout master
git merge better_branch             # fast-forward master up to the merge

If you want your history to be a little clearer, I'd recommend adding some information to the merge commit message to make it clear what you've done. Change the second line to:

git merge --strategy=ours --no-commit master
git commit          # add information to the template merge message

Edit: You didn't say you had pushed to a public repo! That makes a world of difference.

There are two ways, the "dirty" way and the "clean" way. Suppose your branch is named new-master. This is the clean way:

git checkout new-master
git branch -m master old-master
git branch -m new-master master
# And don't do this part.  Just don't.  But if you want to...
# git branch -d --force old-master

This will make the config files change to match the renamed branches.

You can also do it the dirty way, which won't update the config files. This is kind of what goes on under the hood of the above...

mv -i .git/refs/new-master .git/refs/master
git checkout master