I came across similar issue (getting REST API working with TURN server) recently and learned that TURN server doesn't support REST API calls at all and just provides support for an authentication format with shared secret when we enable REST API support in TURN config. The draft only provides info on things that we need to consider while implementing such REST API and WE need to create the API on our own or use something like turnhttp to generate the temporary username password combo.
As @mido detailed, you can implement the username/password generation part in the application itself. But if you have reasons to separate this from the application and want to implement it as an entirely different API service, instead of implementing a complete API as per the draft, I came across another post in which the OP provided a PHP script to generate temp username & password and this one works pretty well once you modify the hash_hmac() function to the following,
$turn_password = hash_hmac('sha1', $turn_user, $secret_key, true);
We need to base64 encode the RAW output of hash_hmac to get it working and I believe this is why it was not working for the OP in that link.
You should be able to test authentication using turnutils_uclient command to verify that the temp username/password combo is working as expected.
turnutils_uclient -y -u GENERATED_USERNAME -w GENERATED_PASSWORD yourturnserver.com
Once you have verified authentication and confirmed that it's working, you can setup webserver for the PHP script to make it available to your application and fetch the temporary username/password combo. Also, you would need to implement other security setup (authentication) to protect the API from unauthorized access.
I know this is an old post, just sharing my findings here hoping that it will be useful for someone someday.