One is to launch a new server and migrate your data from your old server to your new one. Personally that's usually what I do as it means that there is no cruft build up and I can leave my old server running untouched, until I have the new one working. You can use TKLBAM to do that if you wish, or do it manually if you prefer. E.g. WordPress just requires you to migrate the files (/var/www/wordpress) and the DB ('wordpress'). Others might require more poking around.

The other way is to do a Debian style "in place" upgrade. You will also need to download the new TurnKey apt repo signing key for Bullseye. This should work:

codename=bullseye
key_dir=usr/share/keyrings
url=https://raw.githubusercontent.com/turnkeylinux/common/master/overlays/bootstrap_apt
for repo in main security; do
    full_path=$key_dir/tkl-$codename-$repo
    keyring=$full_path.gpg
    keyfile=$full_path.asc
    wget -O /$keyfile $url/$keyfile
    gpg --no-default-keyring --keyring /$keyring --import /$keyfile
    rm /$keyfile
done

The corresponding sources.list (and security.list) lines should like this (respectively):

# grep turnkey /etc/apt/sources.list.d/sources.list 
deb [signed-by=/usr/share/keyrings/tkl-bullseye-main.gpg] http://archive.turnkeylinux.org/debian bullseye main

# grep turnkey /etc/apt/sources.list.d/security.sources.list 
deb [signed-by=/usr/share/keyrings/tkl-bullseye-security.gpg] http://archive.turnkeylinux.org/debian bullseye-security main

Hopefully that info is of value?! Please feel free to ask for further details if you have more questions and/or hit any issues.

Regardless of which path you take, I would encourage you to ensure that you back up everything prior. That way, worst case scenario, you can wind back to a working state.