Why not showing my website name on google search result , showing your hosting provider name..?

Migrating is easy. There are two options you can use:

  1. You can set your custom domain as a Parked Domain so it displays the same contents as your subdomain. For more info, please read:
  1. You can follow the migration guide (which is written for migrating from another host, but can apply in migrating between accounts or even websites in the same account):

If you add the custom domain as an Addon Domain in the same account, you can skip the first 3 steps related to databases, as all databases on an account can be used by all sites on it, so a transfer is not needed.

Redirecting is also easy, you can set up a .htaccess redirect to redirect from your subdomain to your custom domain after you’ve migrated everything to it. There are multiple tutorials on the web on how to do that, like this one: