In addition to the problems from urine and feces and dead horses that other replies have covered there was also a lot of death and injuries from horse and horse-drawn vehicle accidents.

In New York City in 1900 for example the pedestrian fatality rate from horse accidents was higher than the 2003 pedestrian fatality rate from cars [1].

Here's an comment from a medical historian with similar numbers for England and Wales [2]. About 70 road traffic deaths per million population from 1901 to 1905 compared with 80-100 in the 1980s and 1990s from cars. (It is down to around 26 nowadays).

We are better at treating injuries nowadays so it is likely that had they had modern medicine circa 1900 a fair number of those fatalities would have just been serious injuries.

[1] https://legallysociable.com/2012/09/07/figures-more-deaths-p...

[2] https://www.bmj.com/rapid-response/2011/10/31/cars-and-horse...