That Week View is displaying Calendar start from Start of the Week day that you can manually change/configure that for Thunderbird/Lightning Calendar from the following configuration path. But to keeping Today's day at first you should do this everyday!!
Edit menu --> Preferences; under Calendar tab click Views; then change the "Start the week on:" to today's week day!!
But below is a workaround with sed command which it does the same "Change Start the week on:" for today's week day.
sed -n -e '/^user_pref(\"calendar.week.start\", .);/!p' \
-e "$ a\user_pref(\"calendar.week.start\", $(date +%u));" \
-i /home/$USER/.thunderbird/????????.default/prefs.js
Now you can add this script into the Thunderbird's .desktop file to execute it every time you launch Thunderbird in below path.
/usr/share/applications/thunderbird.desktop
Or you can simply run it via crontab everyday at 00:00AM to change it to today's week day! Open to edit with crontab -e in Terminal.
0 0 * * * sed ...
Since this changes Start the week day to Today's Day every-time, so in Calendar it will show the Week start from Today.
Hope it helps!