Look into caching so that it does not need to hit the database for everything. Been years since I used WordPress but there was something like WP Super Cache that worked OK. Your web server (e.g. Nginix) can also be configured to cache pages, and there are entire servers dedicated to caching like Varnish. Be forewarned that caching and cache invalidation can be surprisingly tricky to get perfect.

You also might just actually need a beefier server. But at the very least you should be able to serve the homepage, which doesn't change super often, to everyone.

Congrats on having a site popular enough that you get to learn about a whole new world of tools and technologies!

If it's a wordpress site, I found the best thing for me was to set up Cloudflare in front of the site, because they can cache a lot of requests.

The next best thing is making sure APC is set up on your server. I've also had success tuning settings in Apache to help with memory use, but that's a lot more time consuming.