How can I set the UI language *purely* from the user-account setting for language?

Hello!

I'm building a multilingual website for a mixed community of francophone and anglophone alumni, each of whom have a preference regarding which language they view the website in.
So far, I have been unable to figure out how to make the language of Drupal's UI depend solely on the Language setting in "My account".

For example, I can view the site in either English or French depending on the URL, but if I change my preferred language from English to French (via the "My account" page), the site UI doesn't change to reflect that.

I having considered modify the system so that it automatically redirects a user to the language-correct URL (e.g. switching from English to French would change the page URL from mysite.com/user/1/edit/ to mysite.com/fr/user/1/edit/ , but that seems a bit like a hack and I'd prefer a cleaner solution if possible. It seems like this would be pretty common functionality.

Thanks!