The two Jewish Temples were not located on the current location known as the Temple Mount. That location is where the Roman Fort Antonia was built.
Archaeology, history, and Jewish Rabbis make the case that the Temples were built adjacent to the City of David above the Gihon Spring.
Here is a link to Ernest Martin’s book The Temples that Jerusalem Forgot. I put a link for the Associates for Scriptural Knowledge in the links section on the right.
Here are some videos:
Andrew Gabriel Roth website here.