Zwift is supported on Linux by Lutris.
Lutris is an Open Source gaming platform for Linux. It installs and launches games so you can start playing without the hassle of setting up your games.
The Zwift setup page is here. It says:
After the Zwift installer downloads, you’ll be greeted by a white screen. Quit the launcher, including from the wine system tray
Zwift requires the launcher to be running at all times. Select “Run EXE inside wine prefix” and select “drive_c/Program Files (x86)/Zwift/ZwiftLauncher.exe”. Once it’s running you can hit “Play”.