Google serves search results based on information they have gathered from publicly available source. Due to this, you need to get your information removed from those sources (or have them make it private) before Google Search will remove your information. Of course, there are exceptions to this.
From Google's help on removing personal information:
To contact the website owner:
Contact us link: Find a "Contact us" link or an email address for the site owner. This information is often at the bottom of the site's homepage.
Find contact information using Whois: You can perform a Whois ("who is?") search for the site owner using Google. Go to google.com and search for whois www.example.com. The email address to contact the site owner can often be found under Registrant Email or Administrative Contact.
Contact the site's hosting company: The Whois search result usually includes information about who hosts the website. If you're unable to reach the website owner, try contacting the site's hosting company.
If the website owner has removed the information, it will eventually be removed from Google Search as part of our regular updating process. To speed up the process of removing it from Google Search, use the Outdated Content Removal tool.
Situations Google will take direct action ("personal information that creates significant risks of identity theft, financial fraud, or other specific harms"):
- unwanted & explicit personal images from Google
- involuntary fake pornography from Google
- content about me on sites with exploitative removal practices from Google
- select financial, medical and national ID information from Google
- “doxxing” content - content exposing contact information with an intent to harm
If you have a legal reason for removing data from Google searches, use Google's legal troubleshooter form to get the right results.
Beyond these options, such as in a case where you share a name with someone who posts legal content with which you object, it may be impossible (or prohibitively expensive) to get a search result removed. Be careful before paying a reputation service to do this. There are many unsavory ones out there, so research and consider going with a well-known brand if you don't do the Search Engine Optimization (SEO) yourself.
If you do wish make your search results show up above others', here are some tips. This is not a simple or easy option, but is possible.