Hi @PhilippGerber-6516,
Let's go through Troubleshoot domain and TLS/SSL certificate problems in Azure App Service first so we can nail down the issue you're having. Sounds like you can't validate your host name so let's make sure your TXT records are configured correctly. You'll need a TXT | @ | <app-name>.azurewebsites.net. If you already have that record, try manually verifying your domain. Comment down below none of this worked for you.
With regards to binding the certificate, is managed or 3rd party? Are you using a wildcard or apex? Do you see any errors on the Certificate Binding Operation Detector, under Diagnose and solve problems blade?