Turn up e show up se equivalem no sentido de "aparecer (em algum lugar)", mas, conforme pude ver na Internet, os britânicos preferem turn up e os americanos, show up.
Antes de dar o caso por encerrado, veja o que James B. disse na Internet:
"[...] 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."