I think the main point is that most people are not intelligent enough to use the word "Simulator" in google searches, or similar words. Many people are just kids lookin for a place to play pokemon, so they'll probably just Google "Pokemon" or "Play Pokemon", which unfortunately gives them ROMs, emulators, and the pokemon website. Even wikipedia. -_-.
Addressing the Google searches cited by skylight, I did Google "pokemon", but PS wasn't on the first page for me, which suggests that the search results may differ for everyone. For "pokemon battle", PS's title thingy in the website is "
Pokémon Showdown! battle simulator", meaning if you search Pokemon with any of "Showdown", "battle", or "simulator", most likely PS will show up. Other words like "Play" or "game" may not necessarily give PS as a result.
Completely agree with Arcticblast though - it's pretty hard to advertise a Pokemon simulator website, without fear of retribution and/or drama.
TLDR: If you like PS, and I hope you do, the best way to promote it to others is to pass it on, recommend it, and have fun on it with friends (that's how I got introduced). Tbh that's how word usually spreads really quickly, social media, person to person passing, and not trying to get your link the first result on Google.
Just my two cents.