There was a time when there were no automobiles in New York City, but there was lots of public transportation. (Ok, there were horses and the consequent manure, and the population was way smaller then. But still...)
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If there were traffic laws, nobody knew them and people wouldn't know which way to go to avoid a head on collision.
If you'd just replace cars with horses today it would go a lot different.
And it wasn't just horses. It was horse-drawn carriages. There wasn't that much difference, traffic wise, between carriages and cars. There were traffic jams, out of control vehicles, drunk drivers, street racers, collisions, etc. all back in the 19th century.