This answer focuses on American cities.

Through the 19th century into the automobile age, the vast majority of urban residents did not own a horse. The largest cities had, on average, around one horse for every 23 people at the turn of the century, near the peak of urban horse use. They commuted by walking, by omnibus, by horsecar, or by trolley. Most urban horses were owned by commercial and municipal operators, and were used to haul freight, passengers, and fire apparatuses. In addition, horses in urban environments were rarely ridden (nobody wants to show up to work smelling like horse, or with bow-legs); most individuals who owned a horse used it to hitch a private carriage.

As early as the 1920s, the number of automobiles in cities far exceed the horse-count at its height. St. Louis recorded just under 30,000 horses in 1902; by 1920, it had nearly 50,000 automobiles. Considering that many horse-drawn vehicles require a team of 2 or more horses, it's evident that actual traffic numbers by vehicle would be significantly lower. A survey of traffic at Locust St. in St. Louis in 1885 tabulated 845 vehicles in a day - by 1913, this had risen to 3,496. Similar surveys conducted across the country returned roughly equivalent figures - the lowest traffic increase I have data for is at Broadway in NYC, which saw a meager rise from 1,666 to 3,277 (not even 300% - sad!) vehicles daily between 1885 and 1913.

So, part of the answer to your question is that there was simply less vehicle traffic in cities before the automobile. But what about the horses that did live in cities?

The majority of urban horses Would have been owned by commercial or municipal enterprises, not private individuals, and would live in their own stables. Private horse-ownership was extraordinary expensive: a mid century tandem carriage cost about $1200, and a team of horses cost at least $2000 - higher than the annual income of most engineers and doctors. A horse also required a stable hand to tend it and usually a coachman to drive the carriage, and private stables at home and near work took up valuable urban real estate.

So where did private carriage owners park? Well, by and large, they didn't. A horse is not something that you can just leave unattended in an urban environment: they can be stolen, they can run away, they can bite and kick passers-by. Carriages were recreational vehicles for the most part - commuting in private carriages was reserved for only the wealthiest carriage owners, whose coachmen would pick them up and drop them off at work.

As for some interesting details of urban life before the auto age:

• Transportation cost The growth of intercity and transcontinental railroad networks significantly lowered the cost of shipping freight over long distances, but these savings basically ended at the rail yards and depots. Most cities blocked the extension of freight lines into downtown areas, and prohibited freight traffic on street-level light rail, so terminal delivery was limited to horsedrawn wagons or hand delivery. A 1896 study in Scientific American claimed that intercity freight could be carried at rates as low as $0.25 per ton/mile, while urban freight cost up to $0.25 per pound/mile. The expansion of rail transit sparked a corresponding increase in urban herds, and the widespread adoption of the stronger Percheron breed by teamsters.
• Disease When a major portion of the urban infrastructure relies on living beings, disease can become a major issue. In 1872 an equine epidemic known as the Great Epizootic struck the east coast, killing or sickening horses. That year, the central business district of Boston caught fire - despite state-of-the-art building codes, municipal water supply, and fire department, what began as a localized blaze ended up destroying 776 buildings and $73.5 million of property; the fire department could not get its horse drawn, steam powered apparatuses to the site of the fire. By the time enough day laborers had been recruited to pull the wagons, the department had resorted to dynamiting buildings to control the spread of the conflagration. In other cities struck by the epizootic, sanitation departments could not even haul away the corpses of horses without horses of their own. Epidemics aside, the average working life expectancy for urban horses was just 4 years; by the 1880s, New York was removing 15,000 horse corpses annually.
• Filth According to one estimate, more than one third of arable land in the US was dedicated to raising crops to feed horses - and what goes in, must come out. NYC horses passes between 800,000 and 1,300,000 pounds of manure per day. At the Central Park Stable a 30,000 cubic foot pile of waste accumulated; however, most waste was deposited onto city streets. In the words of one city-dweller, city streets were "literally carpeted with a warm brown matting of horse dropping, smelling to high heaven." More than the aesthetics, horse manure represented a huge public health issue. In dry weather, manure dried and was pulverized into dust by hooves and wheels. Manure contained the tetanus virus, and when manure dust became airborne it provided a vector for respiratory illness - the major killer in urban areas. Although an effective pharmaceutical treatment for TB wasn't developed until the 1940s, mortality rates dropped dramatically after the advent of the automobile.

This information comes from chapter 3 of Down the Asphalt Path: The Automobile and the American City by Clay McShane.