In Denmark, we never have to wait for long. Whenever those in power want to pass something big or important in parliament, they seek a broad support - otherwise, it'll just get repealed whenever they lose power.
Everything certainly isn't perfect here (yes, it can be darn annoying when center parties hold too much power - but then again, it is a stabilizing factor), but healthy mechanics can get you a long way with democracy.
I feel a lot of people in this thread point out cases where democracy doesn't work. That's easy. But maybe it would be better to look for cases where it does work and learn from it.
To take your example of Denmark; Denmark has less population, and half again as much area as Massachusetts. Comparisons between the US as a whole, and individual European countries don't always make the most sense; the more appropriate comparison would be to the EU, to have the same kind of multitudes of disparate people under a common banner.
We have again a situation where 51% crushes the 49%.
Also, in some countries, there are way more than 3 factions (over a dozen in Belgium and the Netherlands), making it less likely that the same subset of parties will find itself in agreement every time.
The effect of having that many factions is that the majority part of parliament will always take the minority opinion into account because they know that, in the next vote, they may be part of that minority.
Of course, that is not a guarantee; I think it is part of a nation's culture. Other countries may value the short-term "let's win this vote" way above the longer term "we have to live in this country together".
That certainly seems the case in the USA, where "in four years time, we might end up with the shorter straw." doesn't seem to play a big role in politics (?anymore?)