Paradoxically means 'in a way that surprises you because it is the opposite of what you would expect.' You use it when two things sit together that feel as if they should not: a very rich person who is, paradoxically, unhappy; a rule that, paradoxically, makes the mess worse. The surprise has a shape — the result is the other way round from the one that seemed obvious. It is a rather careful, written word. You meet it in news, essays, and class more than in chat between friends. It is not a fancy way to say 'very' or 'weirdly.' Strangely only says you did not see it coming; paradoxically says the coming thing was the reverse of the expected one.