Certainly there are many exceptions, but generally those are the stereotypes. How did it become that way in the Western world? Are the any examples of the opposite being true: men with long hair and women with short?
After the French Revolution most men no longer have long hair, fake or real (prior to this men often wear powdered wigs with a little ponytail, this requires a lot of upkeep and became very unpopular after the collapse of the French monarchy). In the 1850-1860's some men might have slightly longer hair like this, but overall nothing too much longer than that in mainstream fashion.
In the British context in the modern era it was part and parcel of the Victorian fascination with all things Classical Rome and Greek, it was a direct imitation of the fashions of the Roman republic when beauty was emphasised for women and dignity was emphasised for men. Long hair was felt to imply time spent caring for and dressing the hair, whilst short hair implied austerity and a focus on less materialistic issues.
The Roman Empire was a general source of inspiration during the height of British Imperialism for reasons that should be self evident, a past civilisation that shares your general outlook is going to be attractive.
You can see these links clearly if you look at 19th c. hair styles, they are often direct copies of the hairstyles found in classical Roman and Greek portraiture.
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On men with short hair, women with long hair, and vice versa
