The answer really depends. Are there growth opportunities in your department? If yes, express interest, perhaps take on some of the responsibilities while in your current role if possible.

Are their growth opportunities outside your dept? Develop a relationship with the manager of other departments, understand what they're looking for. Work to meet those requirements.

Otherwise, consider looking elsewhere at a different company. Don't be afraid to move laterally or take a small step back if there is a strong possibility of growth in the near future. Especially if it's a slight different and promising career direction.

You'll need to convince someone that you can successfully do the next level, figure out who that is (most likely your current manager) and start talking about it.

Do more than they're paying you for and hope someone notices and promotes you.