What does 'turn of the century' mean?

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?

It depends.

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.

See Wikipedia

Both

The turn of a new century refers to the end and beginning of the new century. For your example (1899), it would signal the end of the 19th century and the beginning of the 20th.