Hey, @cfunni!
Ash here from the Shopify team. Thank you for posting your query in the Shopify Community!
I see from the screenshot you shared that your .myshopify.com URL is appearing in search results on Google. I understand that you'd prefer your custom domain to be the one that appears, and I'm happy to help explain why this may be happening, as well as how to fix it.
In order for Google to return the most up to date information on a site, the site in question needs to be indexed on the Google search engine. Since your site is new, it's likely that the search results you're seeing are indexed from before the custom domain was associated with your online store. Now that your custom domain has been added to your account, you can re-submit your site for indexing on Google. Below are some steps for you to take to resolve this:
- First let's make sure your custom domain is set up as your shops primary domain. You can have a number of domains added to a single shop, but all traffic is re-directed to the primary domain. We have a document in our Help Center which goes into more detail on this, and also explains how to set your primary domain. Please check out that document and ensure your custom domain is set as your primary domain before proceeding.
- Once your custom domain is set up as your primary domain, the next step is to find and submit your sitemap. We have a step by step guide on finding and submitting your sitemap in our Help Center for your convenience.
Once your sitemap is submit to Google, it will take some time for their services to crawl your site and index the updated information. Google doesn't guarantee a timeframe on this process, but they do have more information and some troubleshooting steps in the Google Search Console.
I hope this helps, and please don't hesitate to reply back in here if you need further clarification on any of the above points.
