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is the website public?

GTM don't accept you to add attributes in a script tag, it's rewrite it and removes all attributes. Try this format

<script>
  var script = document.createElement('script');
  script.async = true;
  script.setAttribute('data-api-host', '<https://cdn.growthbook.io>');
  script.setAttribute('data-client-key', '{your-client-key}');
  script.src = "<https://cdn.jsdelivr.net/npm/@growthbook/growthbook/dist/bundles/auto.min.js>";
  document.getElementsByTagName('head')[0].appendChild(script);
</script>

Thanks!