To investigate a source using lateral reading literally means reading "to the side". Instead of staying on the page and reading everything on it carefully (reading vertically), you will open multiple tabs in your browser to search for information to provide context about the site itself, organizations, individual names, or topics you encounter.
Open a new tab and enter the [organization's name / website] wikipedia into the search box. What do other sources say about the organization's reputation, funding, and bias?
Another alternative is to use Google's advanced operators to find references to an organization that aren't coming from the organization's own site. (It may still find social media pages or related sites run by the organization.)
Use -site:[domain] to eliminate search results from the domain you specify.
For example: "American Immigration Council" -site:americanimmigrationcouncil.org