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If I haven't mentioned it already, I think the demise of the streetcar was largely due to a perception that it was antiquated. Buses seemed modern and more sophisticated at the time. The ride was also smoother, particularly in Istanbul, where the streetcars had not been updated since the original cars from 1912. As a result, the streetcar network was closed down in the 1960s in favor of buses.
Similar to many North American cities, streetcars have returned to Istanbul. There is the nostalgia streetcar that looks like the ones shown on the postcards, and there are also very modern sleek ones in Istanbul now.
For more detailed information check out the great Wikipedia page for public transport in Istanbul.