Never used sqlite for websites, but wouldn't this problem be better handled by caching? I.e. completely bypass the database, especially for users who will never log in.
E.g. Varnish.
E.g. Varnish.
Having said that, once your site goes truly high-traffic, you'll obviously need some caching. All I'm saying is that PostgreSQL/MySQL can easily handle moderately high traffic if your website is not a heavy CMS.