Edit (October 18, 2024): Ironically the day after this was published, there was a mass migration to BlueSky, with over half a million people joining in a day. However, I don't think they will stick around, and most will go back to Twitter within 3 days.
Ever since Elon Musk took over Twitter, people have been waiting and waiting for a Twitter killer to come. But that probably won't happen for quite some time.
One of the first reasons is time. Most of the social media networks that exist now are popular because they solved an issue early on. However, technology has gotten good enough where we have run out of issues or innovations to really add to networks that will appeal to a wide audience. YouTube became popular because it was video hosting that didn't suck. Instagram made sharing videos on mobile devices quick, while bringing simple image editing to the masses. Twitter launched in 2006, and the main gimmick was that you could send status updates to your friends just by texting 40404.
Twitter was very influential to not just the internet, but the world we know today. You know how people call the pound symbol a hashtag? Twitter. Pull to refresh? Twitter owns the patent to it. The Twitter brand is so iconic that even most people who refer to the platform as "X" still call the posts "Tweets".
Having some racist billionaire buy Twitter isn't going to make most people leave. You have to abandon all your friends, and setup everything again. Just ask Tumblr users. Although Tumblr is nowhere near as popular, the site is still chugging along, although people are still reblogging Shrek memes, so the culture also hasn't really changed either.
I'm betting that the closest thing we will get to a Twitter competitor is Instagram Threads. Threads was SO close to becoming the new Twitter. It launched when Elon introduced rate limits, which made Twitter unusable for a few days. Plus, most people already had an Instagram account, which was all you needed to sign up. You could just import your Instagram followings too. The one thing they got wrong was the algorithm. All you were shown were posts by verified accounts, and there was no "Following" feed, only a for you. This caused Threads to die by day 3, despite having 100 million users. Personally, if they just integrated it into the regular Instagram app instead of making it separate, I think it would still have a chance.
It's hard to talk about Twitter killers without mentioning Mastodon. And my opinion is that Mastodon will never become popular outside of nerd communities. Nerds try pushing it so hard, but there is just so much jank that the average person doesn't want to deal with. Replies are weird across servers, as well as interfaces, and everyone has 4 accounts for the same profile. And even if you can put up with it, all the posts are nerd stuff and art.
Finally, there is Bluesky, the only other network that I think has a chance at competing with Twitter, but it's not there yet. Bluesky was actually created by Jack Dorsey as an experiment or something like that. Jack is no longer apart of the project. Bluesky is basically the exact same as Twitter, but you can make your own algorithms. Also, instead of usernames, everyone has a domain. So instead of me being @wiichicken, I'm @wiggle.monster. The userbase of Bluesky is the same as Mastodon, except there are more normal people, but still mostly nerds. There are 10 million users, but I don't know how many of them are actually active. If you're looking to look at pictures of shirtless furries, whether you want to or not, Bluesky is the place to be.
So... what can you do to find a Twitter alternative? You can wait.