How do you pick a high-growth startup that makes it through several stages? It's like saying the best way to make money is to pick stocks that beat the market.
Look for the phrase "scale-up" instead of "start-up", these are usually well-funded, up-and-running companies that still have some of the start-up mentality but without the financial worries.
Of course, some of these are already a decade old and may employ hundreds or thousands of people already, so YMMV; they might just be like older, boring enterprises, just not as formal or fixed in their ways.
Those are companies that believe they have product market fit though. At that point they are starting to transform from a startup into an established company.
So most of the early stage learnings can no longer happen.
Look at their investors. When interviewing, ask to speak to the investors. See what they say. When speaking with founders do they have numbers they can show or is it pure show-personship?
I mentioned this above, but unlike an investor you don't have to pick a start-up that exits successfully for you to gain really valuable experience that can help you on your own.
Pick a company with a team you respect, a product you connect with, and has money in the bank (just completed a funding round).
You can't go much wrong if you do that, at least in terms of learning.
Of course, some of these are already a decade old and may employ hundreds or thousands of people already, so YMMV; they might just be like older, boring enterprises, just not as formal or fixed in their ways.