to show up = "to arrive somewhere in order to join a group of people, especially late or unexpectedly".
Given the 'unexpected" quality of "showing up", it's less likely to be used in the first person describing a future event as a certainty.
• A possible future event: If I show up, they'll be surprised.
In AmE "show up/turn up" usually implies the unexpected.
"Drop by" is a quick visit, but I don't think that it necessarily implies an unexpected visit, just an informal one.
I would never just "show up" at a friend's house, but I would "drop by"... giving notice that I plan to do so.
Both are used in AmE but have different uses.
There tends to be a sense of agency associated with "show up". Ex: I showed up late for work because of traffic.
"Turn up" is usually associated with objects. In films and murder mysteries, a missing person may "turn up" dead or a piece of evidence may "turn up" unexpectedly.
Corpus of English: AmE 2009 https://tinyurl.com/y3vzp3k5