After a new website goes live, we need to submit the website to search engine like google, bing.com, yahoo.com or duckduckgo.com
Use keywords or Key Phrase to check to on search engine whether the website has been listed is not that accurate.
For better checking, we can using Google Search Bar or Bing.com Search Bar to check.
Example: In the Google search bar we can type site://www.entertopseo.com.my
If the website not yet been indexed, you search result will show
"Your search - site:abcdefg.com.my - did not match any documents. "
If the website has already been indexed, you will see its pages in the Google search results
The next step you should go for mobile friendly test for your website.
Need a SEO Specialist to help you to get indexing and ranking data from Search Engine.