When we search "health insurance" keyword on search engine Google, website (apollomunichinsurance.com) shows the title "Apollo Munich Health Insurance" on the SERPs. It should show the original title which is "Health Insurance India, Medical Insurance India, Health Insurance Policy".
What's wrong? Something wrong with the website? What's the solution in such cases??
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The current title you have in the title tags appears a bit spammy to me, which is what I'm guessing is going on here. Whenever your title doesn't match the content on the page or if googlebot thinks you're stuffing the title with keywords it will choose to ignore the title.
Specifically, Google notes that:
If we’ve detected that a particular result has one of the above issues with its title, we may try to generate an improved title from anchors, on-page text, or other sources. Webmaster Answers
Breaking down the title you have 2 instances of "Health", 3 instances of "insurance", and 2 instances of "India" within the current tile "Health Insurance India, Medical Insurance India, Health Insurance Policy"
If I were you I'd try changing the title to something like "Apollo Munich Health - Health Insurance in India". A title like this likely won't be picked up as spam and should correct the issue.