They aren't compatible. What's more - they never have been.
Here, let me show you true democracy:

Democracy is the tool of a majority to enforce their will upon a minority. It gives an illusion of "say in the decision", but ultimately those who control the votes control the outcome.
Freedom is about a lack of encroachment of others into one's own affairs. This means that - for example - freedom is about a majority not being able to tell you "you can't marry this person", or "you can't smoke this plant", or "you can't own this gun". It also limits your ability to encroach on others, which is why you have no freedom to, say, walk up to someone and punch them in the side of the head.
Democracy on the other hand is the opposite of that, it is a majority of folks saying, "We don't care if you have made a decision for yourself, you're not allowed to poison yourself with that plant," or "We don't care if both of you are in love and happy, we say you can't be joined together in marriage," or "We don't care if you're not actually harming someone with that gun, we say you aren't allowed to have it."
So I can't pretend to know what Peter Thiel was thinking about with that quote, but that's the sense that it evokes in me. I can't, in fact, understand how anyone ever consider them compatible.
When markets turn after a long trend then shortly before and after the extreme funny things happen. Gordon Brown sells our gold reserves at the low, because what's the point of owning gold. UBS fire Tony Dye, notorious tech bear, just as the bubble begins to burst. Central bankers in 2007 have an outbreak of self congratulation on having single handedly achieved the reduction in economic volatility termed the great moderation just before volatility explodes. And what's true of markets applies to other social phenomena too.
It's inescapable if you are involved in such domains. It's okay to succumb temporarily, provided you don't stubbornly fight it and quickly realize your mistake.
I don't know, but I think that's what happened to Thiel - a moment of despair before the dawn.