A rear-steering car is an example of an inverted pendulum, which as many have mentioned is inherently unstable and infamously difficult to control. Using this design will make cars more dangerous, will require roadways to be significantly wider, and will limit how quickly your vehicle can turn. If you really want this in your world, though, here's a way you can contrive it into being:
Henry Ford didn't return home when he was 19.
At age 16, Ford left home to work as an apprentice machinist for several companies in Detroit. He returned to the family farm three years later, where he worked for almost 10 years on a farm and a sawmill until he became an engineer for the Edison company. It was here that he developed his quadricycle, the precursor to his automobile.
In your world, Ford never returned home. Instead, he continued to work in Detroit, where his last employer was Dry Dock Engine Works, a manufacturer of engines for boats. Ford stayed here longer in your world, where he advanced to become the chief test pilot and analyst for new engine designs. He spent many, many days piloting small craft around the Detroit River, and he particularly enjoyed the tugboat that they used for testing cargo hauling capability. As a result, his automobile designs did not evolve from the horse-drawn buggy. Instead, he took a wildly-successful tugboat design and modified it to operate on land.
The resulting vehicle would be built in a radically different manner. Your cars would look more like this photo of a tugboat pushing a barge:
The driver would sit in a small cab, most likely alone, with the engine and controls. This cab would link up to a payload of some sort, whether that be a passenger compartment, flatbed for cargo, etc. and would push it along. The payload section's basic frame would be structured somewhat like of a train car: a flat platform with its own fixed wheels, but dependent on external propulsion and steering.
All of the complex, expensive machinery is encapsulated within the cab. Customers could buy a single cab, plus several relatively-inexpensive payload units which would allow them to easily haul cargo, take their family to town, or even do light farm or industrial work without having to buy separate vehicles. A delivery driver could drive a box truck, a liquid tanker, and a refrigerated food van in a single day, all without leaving his cab. Ford was also known for his standardized designs, which allowed customers to upgrade to a newer cab design while still using their existing payload units.
Engineers would undoubtedly notice the inherent stability problems with this arrangement once speeds increased to a certain point. However, Ford's design was ubiquitous and deeply entrenched by that time, and the existing ecosystem of cabs and payload units meant that redesigning the vehicle for higher speeds would require customers to abandon their extensive investments in it. That would make the new designs far too expensive for all but the richest customers, not to mention you'd have to re-train every driver on the road. Drivers were even frightened at the very thought of piloting a vehicle from the front. The payload unit offered good protection in the event of a collision, and moving the cab to the front presented a lot more danger for the driver and for the expensive part of the vehicle. Instead, the existing design stuck, and vehicles in your world have inherent limits to their speed and maneuverability that drivers simply accept as a normal thing.