I'm European too and I really wish people from Europe didn't have an attitude like yours. Some parts of Europe are incredibly poor, but of course we have a much smaller homeless problem, because if you're truly without shelter then you simply die in winter.
I'm from Spain and there are homeless people in Spain, just a lot less than in the US. But it's not because they die in winter, it is because you don't become automatically poor if you lose your job, or if you need an operation, or if you study at university. It's safety nets that avoid people losing everything and becoming homeless.