This is my first answer on AskHistorians, so if anything doesn't follow the rules, I apologize.

1. The U.S.'s cities tend to be new in comparison with, say, European cities. For example, contrast London, founded in Roman times, with, say, Chicago, incorporated as a city in 1837. Cities founded before approximately the Renaissance (founded by Europeans) tend to have very narrow, winding streets because of the way such cities grew, incrementally. In contrast, cities founded after tended to be planned, and planned cities usually have wider streets to combat problems like lack of open space and accommodating carts. It has been theorized that early Americans, living in a less dense country and often having escaped from dense cities, particularly enjoyed the open streets such plans demarcated. See, for example, the Commissioner's Plan of 1811 in Manhattan, which long before the invention of cars outlined Manhattan's very wide avenues. There are exceptions to this tendency, but they tend to "prove the rule": Paris's wide boulevards were actually created in the 1800's, and Boston's narrow, winding streets were mostly established piecemeal in colonial times. For these reasons, American cities were usually well-suited for the automobile by already offering the wide spaces needed for cars.

2. Railroads in the US have always tended to be private companies, while railroads elsewhere are more likely to be public or nationally linked. In the US, the popularity of the "robber baron" stereotype of railroad moguls (not entirely unwarranted) contributed to an unfavorable image of the railroads. Roads at the time were partially seen as providing an alternative to the greedy railroads: instead of a farmer being forced to use his local railroad, which might charge him exorbitant amounts for shipping, he had the option of trucking his produce to a more competitive company. This helped to spur early road development in America. In Europe, roads had been established for thousands of years, and railroads were mostly national, so there was never as much of an incentive to build a large road system.

3. Because they were private, railroads never enjoyed the considerable federal subsidies for interstate construction in the 1950's and 1960's. Big cities, desperate for infrastructure, often took whatever money they got to build interstates right through their centers, paid for by the federal government at 90 cents to the dollar. In addition, the interstates were often used as a slum-clearing method: instead of cleaning up a poor area, the cities would just build freeways through them, destroying miles of urban fabric. The hollowing out of city centers that occurred further incentivized the sprawl enabled by cars, creating a feedback loop encouraging a total conversion of cities to auto-based transportation. European countries never embarked on a project as ambitious as the US Interstate System, partially because WWII drained these countries' resources. (The closest system to the US's, Germany's Autobahns, never ran through central cities). With more intact city cores, railroads never suffered the same competitive disadvantages as they did in the US, and the cycle of car-dependence never took hold in Europe as it did in America.

Sources: Suburban Nation, by Andres Duany, Elizabeth Plater-Zyberk, and Jeff Speck contains a short history of America's auto dependence. Many of the resources at the Federal Highway Administration, such as this history of the Interstate system, are great introductions to the history of transportation in the United States. The Death and Life of Great American Cities by Jane Jacobs is the foundational work on how auto-dependence changes cities.

Your question seems to indicate that the US didn't provide significant assistance in the developments in rail but the US government provided significant subsidies to railways in the 19th century.

Four of the five transcontinental railroads were built with assistance from the federal government through land grants. Receiving millions of acres of public lands from Congress, the railroads were assured land on which to lay the tracks and land to sell, the proceeds of which helped companies finance the construction of their railroads.

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What subsidy to cars are you referring to? For most of the history of the US highway system, it was largely, if not entirely, paid for by fuel taxes and other user fees that would be paid solely by motorists, because using general taxes to build highways for wealthy motorists was politically unthinkable. Most exceptions were justified as jobs programs (during the Depression) or as aid to farmers getting crops to market.

One cause for the priority of cars was the development of suburbs and housing for all which made it necessary to own a car. Suburbs didn't got a link to a public transportation system. European cities never developed a similar structure. In countries like Germany it's common to rent a apartment or house.

Another difference to Europe was, European public transports were preferred because after WW2 almost no worker was able to buy a car and this changed first at the end of 1960s.

There is a massive difference about urban planning between Europe and the US too. Cities in Europe are trying to develop a long term plan maintaining ideals, like preserving the structure of a downtown or a strict separation between business and living areas.

While not the only reason, one of the big reasons was the growth of the US automobile industry during the 50's and 60's when little competition existed. Remember this is the age of the suburb, and suburbs are not conducive to expensive, dense public transit systems. Also, as others have pointed out, streetcars disappeared everywhere, although public transit itself did not disappear to this great extent outside the US, and to some extent Canada. Part of the reason also being density.