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When looking for official country, state or city websites, a good rule of thumb is to limit your domain (within the advanced search) to the appropriate abbreviation.
For country sites, refer to a list of country-specific domains. Add your topic to Google and add site:fr (if France is the country of choice).
For state sites, limit the domain to state.state abbreviation.us (i.e. state.ny.us). Add your topic to Google and add site:state.ny.us.
For county/city/town sites, the process isn't as cut and dry. Try adding your topic, the name of the city of interest and limit to the appropriate state. For example, a Google search might be health Rochester site:ny.us. Some cities/towns will list their websites under ci.name of town.ny.us (i.e. ci.webster.ny.us). Others will use a .org (i.e. geneseony.org, townofbrighton.org, cityofchicago.org). Even still, cities may refer to the broader county is which they're located (i.e. co.monroe.ny.us, co.livingston.state.ny.us).