How can a painfully introverted person increase their visibility and promote themselves?

Great question! I'm more of an introvert myself, and my shyness was really extreme all the way back to college. I totally resonate with a lot of the material here.

My overall advice here is to start with the smallest audience possible and then work your way up.

1. The first step is to get comfortable sharing your wins at all. As an intern at a well-run, giant tech company like Google, I assume you have a dedicated mentor who you have recurring 1 on 1 meetings with (that's how it worked at Meta). Share your wins in those meetings! It may feel weird, but the entire point of an effective 1 on 1 meeting is that they should feel awkward. There's a good chance that your mentor will say something like "Oh wow, that's awesome - You should share it with the rest of the team!", and voila, you now have a "green check" to share more broadly.
2. Once you're comfortable sharing wins in a 1 on 1 setting, try sharing in a team setting. If your team does daily stand-up, that's a relaxed, warm environment to do this. Some teams may not do standup, but they'll generally have a weekly/bi-weekly meeting at least - That's another avenue. For async methods, there's Slack and Workplace (I imagine Google doesn't use these though; hopefully someone who's worked at Google can share the correct tools here 😅). Tactically, here's a bunch of advice on how to write a great project update.
3. Sharing at a team-level is generally enough for an intern, but I recommend going even broader if that's possible (you can talk to your mentor about this). If you can share at an org level (i.e. everyone under your director), that's super cool. And for maximum visibility, you can do LinkedIn, especially if it's a public-facing launched feature.

Zooming out, teams generally love interns and spend a lot of energy caring for them - You're the next generation! I know that this is easier said than done, but a lot of this is taking that leap of faith and just putting your foot out there. You'll be surprised at how warm and supportive people will be of that, and from there, you'll want to keep sharing more and more. The tricky part is getting started.

Lastly, if you still have anxiety with all this, I'm happy to chat! Just reach out to me on the Taro Premium Slack, and I'm happy to book some 1 on 1 time for us!