The other answer is fine, but, I'd add that "turn up" can be used for non-physical things while "show up" cannot. "We thought everything was going to be fine, but some problems turned up" makes sense but using "showed up" in that sentence would not. Also, I don't think it is an American/UK thing. Here in the US we use both quite often.

The major difference is that "turn up" is UK's style of speaking, and "show up" is American. The other difference is that "turn up" has an element of being unexpected. Being such, we can say, "After discovering the trails to the shack, the bodies we were looking for turned up dead." You wouldn't say "showed up dead." "Show up" seems to have a "will act" feel. To turn up, however, seems to have a more passive feel (which is why something that is dead turns up dead instead of shows up dead, being that showing up is not an action done by something dead).