Here's why:
* Turn of the century specifically refers to the transition between one century and the next. It encompasses the final years of the 19th century and the initial years of the 20th century.
* End of the century focuses only on the last years of the 19th century.
* Beginning of the century focuses only on the first years of the 20th century.
While both "end of the century" and "beginning of the century" are valid phrases, "turn of the century" is the most accurate and inclusive term for the transition period.