What race to Mars? At best three entities are even aiming to send people to Mars, and all of them at a quite leisurely pace: NASA, ESA and SpaceX. And by all accounts, they are cooperating more rather than competing.
Some people take Elon Musk too seriously. His timelines are always overambitious. Starship will not be going to Mars (even unmanned) any time soon. Even though Musk says it's necessary for interplanetary travel, at least the first dozen or "Starships" or so will have much less ambitious tasks: Launching Starlink satellites, maybe other commercial payloads and of course going to the moon. All of those missions require multiple prototypes. I'd be surprised if any Starship makes its way to Mars in the next five years. More like ten years, even for the first unmanned mission.