You can't.
Social media is public, period.
If you want private social interactions then use end-to-end encrypted communication services.
Any oppressive regime will have multiple tools that can identify your access to social media services and determine who the human is behind the keyboard. Assume that everything you post on public social media has your real life name and address attached to it.
Unless you're an expert and taking extreme measures (and you wouldn't be asking this question if you were such an expert) then anonymous access to online services with respect to a sovereign power with an intelligence service is impossible.