Why aren't jet engines used in cars? From my "research", I found that there have been efforts to make cars with jet engines in them (mainly race cars), but all of them relied on the thrust produced by the jet engine.
Why can't a jet engine be used to turn a driveshaft the same way an ICE turns a driveshaft?
This would mean not making use of any thrust generated by the jet engine and only leveraging the rotational force produced by the jet engine.
 A: One thing that hasn't been mentioned yet is that although gas turbine engines have an excellent power to weight ratio, they are less efficient than piston engines of similar power output, especially in small sizes.
Gas turbines typically use only a third of the oxygen in the inlet air, or to put it another way, they ingest three times the air required to burn the fuel. This is because the hot parts of the gas turbine (especially the turbine blades) are subject to constant high temperature and would melt if the minimum air required for combustion was used. On the other hand, gasoline engines use the minimum air required for combustion (as do diesel engines when operating at full throttle.) The combustion temperature in a piston engine is therefore higher than in a gas turbine, but it does not damage the cylinder walls because they see only the average temperature of the full cycle.
The low flame temperature in a gas turbine limits the theoretical Carnot efficency. Because the gas turbine uses three times the mass of air required for combustion, a lot of energy is carried away in the exhaust. A more detailed analysis will show that these two statements are more or less equivalent.
Thus a truck diesel engine can achieve 50% efficiency (with the help of a turbocharger) which is unheard of in a gas turbine. Combined cycle power stations achieve better than 50% efficiency by raising steam from the exhaust gas of gas turbines to extract extra energy. A typical mass balance is: 100MW fuel, 40MW shaft power from the gas turbine, 40MW steam raised from the exhaust, 20MW lost to the chimney. The steam raised can then be used to generate electricity at 40% efficiency, yielding another 16MW. Total 56MW electric from 100MW fuel.
Installing a combined cycle plant in a car (or even a truck) is prohibitively complex. Therefore, piston engines offer better efficiency.
This is especially the case in small sizes. Better efficiency is achieved by high combustion chamber pressure. But the pressure that can be developed / recovered by a single row of blades is proportional to the square of the blade velocity, which means that rotors in small gas turbines have to spin very fast or have many rows of blades on them in order to achieve decent combustion chamber pressure.
A: Here is a short list of the reasons why.
To serve well on roads, an engine must be capable of a wide range of operating speeds from idle (near zero power) to max power. Turboshaft engines "idle" at somewhere around 60% of max power and reach peak operating efficiency at near 100% power. This makes them unsuitable for stop-and-go operating conditions.
Ideally, a car engine would be sized in the 100 to 200 HP (75–150 KW) output range. This could be done by scaling down a turboshaft engine but efficiency suffers significantly below ~600 shaft horsepower (450 KW). The fuel burn rate for mini-gas turbines in general is really awful.
Turboshafts produce a very energetic plume of very hot exhaust gas which would remove the paint from the car behind you when stopped in traffic.
Finally, turboshaft engines are best for applications where their excellent power-to-weight ratio justifies their (high) manufacturing costs. This is why they are popular in corporate aircraft and luxury helicopters but decidedly not in economy cars.
A: A little more information.
Chrysler did build a protype commercial jet engine car in 1963-1964.  Search the web for Chrysler Turbine Car for more details.  One problem is that leaded gasoline (the most common gas at the time) left damaging deposits on the engine. Other problems were slow acceleration, sub-par fuel economy, and relatively high noise level. Advantages of the Chrysler car were its smooth and vibration-free operation, reduced maintenance requirements, and ease of starting in different conditions.
A: To add a bit of nomenclature to the other answers, gas turbines get different names depending on where most of the power goes:

*

*in the exhaust, it's a turbojet (jet aircrafts proper)

*to drive a shaft, it's a turboshaft (helicopters, trains, other vehicles)

*to drive a propeller, it's a turboprop (aircrafts)

*to drive a fan, it's a turbofan (modern airliners)

So this has been done, but under a different name.
A: 
My question, then, is this: Why can't a jet engine be used to turn a driveshaft the same way an ICE turns a driveshaft?

These exist. Though the branding is different. They are not called "jet" engine but gas turbine engines.
Actually, "jet" engines like you see on a passenger jetliner or military jets are called "gas turbine" engines. But since the word "jet" became trendy they are more popularly called "jet" engines and are also branded as such (for example their fuel is branded as Jet-A and Jet-A1).
There's a company called Designline that makes gas turbine buses currently in operation in 5 cities worldwide. There's another company in Italy called Brescia that is doing something a bit different. They're using microturbines (kind of like the jet engines you see in model airplanes and that flying wing jetpack and flying iron-man suit you see on youtube) to generate electricity to recharge batteries and drive electric motors.
Also famously the American main battle tank the M1 Abrams (and its variants) use a large gas turbine engine.
However gas turbine engines in land vehicles are not without problems. For buses there were several companies that did not survive trying to bring their products to market. Most of them suffered huge losses covering warranties or cancelled orders due to reliability issues. The maintenance turned out to be very expensive. It looks like Designline has managed to solve a lot of the issues and are doing quite well.
The same was true for the M1 Abrams tank. There were reliability issues in the early variants. But because the client is the US Army and they really needed the horsepower and they could financially afford it they stuck with it and resolved all the issues in later variants.
So that's one reason why companies don't manufacture gas turbine cars. It's the same reason nobody wanted to build an electric car until Tesla proved that it's profitable: there's a huge financial risk to bringing a new engine type to market. Even electronic ignition in piston engines took a very long time to become mainstream and the Germans were using them in WW2.
A: 
My question, then, is this: Why can't a jet engine be used to turn a driveshaft the same way an ICE turns a driveshaft?

I will try to add some reasoning about the answer a little too long to be a comment.
First point is that every technical "device" is a compromise of a number of factors. Some of the factors are technological, some are related to manufacturing, many are related to market factors (ie what will the buyers buy).
So let us look on cars. The typical US car is (too) big, guzzles (too much) gas or diesel as well as quite expensive. From that point of view it would be possible to use a gas turbine in the current car format. The size of the car could take it after a bit of redesign, the fuel is readily available from lots of places. But apart from some special usages and one-off enthusiast builds it has not happened. Some military vehicles does use gas turbines, and I believe you can find one or more usages here and there. And of course, you could build one yourself as a hobby project.
There are a few different hurdles to pass for gas turbines to become used in "general cars" found on our streets. I am no expert but here are some of my guesses:

*

*It is technological challenge to transfer the torque from the gas turbine to the wheels. It could probably be done, my hunch though is to go with an electrical transmission, ie a generator and electrical motors. My guess though. Cars typically need a lot of torque quickly available when accelerating and idles down to a lot less torque needed when cruising. The max torque would be the design factor in "direct drive" but could be handled by adding energy storing units (batteries?) in the electrical drive scenario.

*The cost factor. Small gas turbines are available but they are quite costly. From a sector I know better, General Aviation, a new gas turbine costs 10 times or more compared to a traditional engine. My hunch is that it would be similar for car engines, at least initially. The cost is of course to a large part due to volumes, traditional car engines are made in specialized factories at quite low cost per unit, as the investment in design and machinery can be amortized over a lot of engines. Add to this that you would need to train a lot of mechanics and give them new tools in order to service the engines, not without cost.

*The efficiency factor. My hunch is that gas turbines in a vehicles would use more fuel than a traditional engine. I might be wrong, but our current gas or diesel engines are very efficient and highly developed.

*The environment effect. Gas engines as well as diesel engines has special additions to decrease amount of gasses impacting the environment. Gas engines does this using a catalysator, diesel engines through add-blue. Something similar could probably  be done for gas turbines. My guess though is that the way forward there would be to use alternative fuels, maybe hydrogen or perhaps alcohol (not an expert here, it might not give the wanted results).

*The electrical future. It seems like the world currently is moving towards electrical vehicles. In this market, adding yet another gas or diesel engine technology would seem very risky from a marketing point of view. It would require quite an investment into new toolings and rely on beeing able to sell large volumes which seems not very probable.

*And the consumer: what is the unique selling point for a typical user commuting to and from work? It escapes me.

Of course, for special usage and enthusiast motors it may very well happen. I guess, a gas turbine Ferrari or Lamborghini might be able to carry the costs and perhaps be seen as a good "status" marker.
So in conclusion: it simply will not happen soon.
A: They are used. That's what "turbochargers" are.
Simply put, they are used in automotive engines, as a mechanism for improving the power of the piston engine. The exhaust from the engine is directed into a turbine that then burns any leftover fuel and extracts any waste heat left in the exhaust, and then uses that energy to drive a turbine to draw in more air for the piston engine, in order to increase its power and efficiency. We call such jet engines "turbochargers".
If you're referring to a jet engine being used as the primary source of locomotive power for the vehicle, rather than as an add-on to boost the capabilities of a piston engine, I will direct you to the M1 Abrams battle tank, which uses a gas turbine engine. This engine was chosen instead of a conventional piston engine because it provided enough power to drive such a large and heavy vehicle at high speeds, at the expense of high fuel consumption. Since heavy civilian vehicles such as long-distance trucks are usually much more concerned with fuel efficiency than performance, they're incentivized to use more fuel-efficient diesel engines instead.
