I use the Widget language switcher. When I am logged in and see the home page the flags appear at the correct size.
When I am logged out ..as anyone would see my site the flags have an other size.
Could you please share temporary access (WP-Admin and FTP) to your Website? Since I can't log in to compare the difference, I won't be able to help much here.
IMPORTANT: Please, note, a backup is required for security reasons, and to avoid any data loss, you can use the https://wordpress.org/plugins/updraftplus/ plugin for these backup needs.
Your following answer will be private, which means only you and I have access to it.
✙ I would need your permission to de-activate and re-activate Plugins and the Theme and to change configurations on the site if required. This is also a reason the backup is critical.
Furthermore, I’ve checked your debug information and I noticed that your “WP Memory Limit” PHP limit is below the required 128MB or recommended 256MB as described at: https://wpml.org/home/minimum-requirements/ . Could you please contact your hosting provider and ask them to increase this PHP value for you and also, ask them to update the MySQL version to 5.6+?