If I wanted to write about 1899, would I call it the turn of the 19th century or the turn of the 20th century? Basically: does 'turn of the century' refer to the beginning or end of a century?
In British English, according to Wikipedia, you would call it the turn of the 20th century. The turn of the 20th century includes the end of the 19th century and the beginning of the 20th.
In American English it can be as above, but it could also be referred to as the turn of the 19th century, i.e. viewing it as the century turning from the 19th century rather than as turning to(wards) the 20th.
Generally speaking, the turn of the century refers to any century where the turn is occurring.
Context will tell which century it is.
Imagine, people in the year 999 experienced the turn of their century as being from the year 999 to the year 1000. The actual turn is ineffable.
Basically, it refers to the point where one century becomes another. Just like: the turning the corner: you go from one street to another.
When I turned the corner, I realized I was in Harley Street.
Here's Wikipedia, see the first part only for my answer:
For me, the turn means the moment it changes INTO the century that follows. And I just do not agree with Chicago Manual of Style, insofar as "n-th century" means the same in BrE and AmE. I mean specifically that n-th century means the same in both,ergo, the turn must then refer to the one you arrive at after the turn. I do not believe there is any discrepancy in BrE and AmE for 19th century or 20th century.
[correction made: "insofar as the turn of the n-th century" was a mistake]