White House.gov is the website of the current President and his Administration. It reflects the issues, concerns, and priorities of the White House today. Students can use this site to locate current issues, as well as research previous presidents through the "About the White House" portal.
Because each President's website is the voice of its administration, websites are archived in the same way other presidential records are: captured and 'frozen' as the new administration comes on board and kept for archival research and exploration. Links are not kept up, but the individual pages are kept as they were. The presidential websites of Clinton, Bush, Obama, and Trump are archived here: https://www.archives.gov/presidential-libraries/archived-websites
To learn more about the presidency and engage your students with history - start their explorations as Advisors to the President. Enjoy!