Why, in old movies and TV series, do they always use such extremely exaggerated wheel turns when driving a car?

Other answers are correct when mentioning the absence of power steering at the time and the fact that the actor/actress ‘driving’ could not see the screen behind them, but budget and time constraints were also an important factor as was, in some cases at least, cinematic technique. Note also that exaggerated physical movement by actors could make up for a lack of camera movement to prevent scenes from appearing to static (we are talking about motion pictures, after all), and that it can serve a dramatic purpose (as Mark C. Wallace observed in a comment).

Note: This answer owes much to a recent discussion I had with the actor Barry Truex, who was active in film and TV series in the 1950s and early 1960s. Specific points made by Mr. Truex are followed by (BT); his observations have greatly enriched this answer, for which I would like to thank him.

One initial point that needs making is that, looking at a selection of clips from the 1940s and 1950s, the amount of steering wheel movement used by actors and actresses varies significantly – some move the wheel a lot, others very little, but still more than seems natural today - see, for example, clips here, here and here. The reason for this variation is at least partly due to budget (on which more below).

Power steering, requiring less movement and less force than manual, was not standard or an option on all American cars until the late 1950s / early 1960s, so the vehicles used in films would mostly not have had it, especially on low-budget productions. Further, actors pretending to do some everyday activity naturally draw on real-life experience, and few actors would have had more experience with power steering than manual, so manual is the ‘style’ they used. Note also that the vehicle on the set was sometimes no more than a prop so the steering wheel may have had little resistance, making it difficult for actors used to some resistance with a real car.

Generally, no one instructed performers on how to steer – they just did it, especially on low budget productions operating on tight schedules where the director had little time for fine-tuning scenes (BT). On big budget or prestige films, however, more attention was given to such details and ‘exaggeration’ is generally less noticeable, but it is important to note that filming the wheel movement used today (with power steering) would not have seemed natural to viewers in the 1950s.

"The Tuttles of Tahiti" (1942) rear projection. Source: Hollywood Historic Photos

A more challenging issue was the use of rear screen projection which the 'driver' could not see. This technique became widely used from the early 1930s when most films became studio-bound.

While filming in moving vehicles was not unusual during the silent era, sound technology had made it all but impossible due to bulky microphones, wiring, and larger film cameras requiring soundproof “blimps.”

Even for major studio productions, it simply didn’t make sense economically to go on location to film a one or two minute segment of two people talking in a moving car (BT), and there were the additional problems of maintaining sound quality and the (at the time) limited technology for filming while moving with the vehicle. Nonetheless, alternatives were used at times: in Orson Welles' Touch of Evil (1958), cinematographer Russell Metty mounted a camera on the hood, using the back seat for the batteries and wires.

There was also the option of filming from behind, but this had obvious dramatic limitations. Consequently,

the practical benefits involved in rear projection — speed, efficiency, consistency, and controllability — largely explain why it was the favored technology for so long.

Source: Turnock

Source: crimethrillercinema

Further,

Research and reviews suggest that rear projection was not a major problem for directors, critics, or the mythical “average” moviegoer, and the studio technicians thus found their aesthetic standards to be out of sync with the majority, yet they continued working to improve the technology.

Source: Julie Turnock, 'The Screen on the Set: The Problem of Classical-Studio Rear Projection'. In 'Cinema Journal', Vol. 51, No. 2 (Winter 2012), pp. 157-162

Stuck for the most part with rear screen projection, producers, directors and performers did the best they could, considering the technological, time and financial constraints imposed upon them. Not being able to see the screen, ‘drivers’ had to improvise. Sometimes the end result was hopelessly out-of-sync, especially when the ‘driver’ turned left while the rear screen was heading in the opposite direction. Directors with a little time and money to spare could reshoot the worst bits, others couldn’t (BT). Among other problems adding to and sometimes magnifying the deficiencies of the ‘driver’ were focus and depth; unless the film being projected onto the rear screen was very high quality, it looked obviously grainy, unfocused and weak compared to the actors in the car.

Auto-cutout Stock units "As Good As Married with Walter". Source: Hollywood Historic Photos

In general, quality was often compromised because the demand for product from TV networks was huge so production costs had to be kept low for the TV stations to afford new programming. The studios, still reeling from the effects of the antitrust cases of the late 1940s, struggled to keep up. Each episode of a 1950s TV series often had to be completed in three days; there was no time for actors to study the footage that would playing behind them before attempting to shoot the scene (BT).

Nor was there the time or budget for reshoots or filming short segments on location in place of rear screen projection (BT). Consequently, the poorly synced scenes stayed in. Actors, actresses, directors, film technicians etc. in the 1950s were well aware that the effects were often poor, sometimes laughably so, and it was commented on, but the aforementioned constraints meant that it was rare for any changes to made (BT).

Other sources:

David Puttnam, ‘The Undeclared War: The Struggle for Control of the World’s Film Industry’ (1997)

Robert J Read, 'A Squalid-Looking Place: Poverty Row Films of the 1930s' (PhD thesis, August 2010)

Birk Weiberg, 'Image as Collective: AHistory of Optical Effects inHollywood’s Studio System' (PhD thesis, 2016)

turning the wheel wildly to the left and right

For the very simple reason that they could not see the images. So you get them to make random motions, because steering straight while the scenery is moving about is even weirder, your brain is telling you the car is not moving because they're not doing anything.

In the 1950s the typical technique was to put the car in a studio and project the background on a high-quality screen behind the car. Careful synching was required so this didn't cause flicker. Naturally, the actors can't see the imagery that's behind them and thus can't synch to its motion.

One might imagine that you could project in front as well, perhaps to a smaller screen. But any projection that's bright enough for them to make out while bathed in kliegs is bright enough to get picked up on the camera, so you would see the images, or bits of them, reflecting off the actors and frame of the car and you would get serious flickering.

This was all improved when blue screen techniques became cheaper, so you could add the motion after filming. That had the same problem cueing the actors, however, as now there's no image at all.

If you wanted to get really fancy, you could do what Kubrick did in 2001: A Space Odyssey and film the entire sequence with the actors looking at a screen in front of them that is synched to the camera, basically the reverse of what they did originally to film such things. Then you develop that film and front-project it onto a screen in the window the car. Mucho Denaro!! Looks awesome in the end though, check out the bit where you can see the pilots of the Aries in the (red) windows as it lands on the Moon. Roughly around this point in the film.

Where realism was required, you put the car on a trailer and tow it along. The actors can then easily follow the motion of the car with realistic wheel inputs. You don't even have to tell them to do it, it's entirely natural. Michael Douglas went sailing by me in such a setup some time ago.

In addition to the points that Maury Markowitz described, another factor was the large amount of free play in many car steering systems of that era.

I was a young kid in the late 1950s, and I distinctly remember getting to sit in the driver's seat of various family cars (while parked, of course) and "drive" them with my hands on the steering wheel. Unlike a modern car, you could freely move the wheel back and forth a good inch or two without moving the front tires at all!

This wasn't because of the lack of power steering, it was just slop in the recirculating ball steering system.

You could feel a bump when you reached the end of the free play and if you turned the wheel really hard, you could move the front wheels a bit. But that wasn't much fun, it was easier to just move back and forth in the "dead zone".

Imagine driving a car like that on a moderately windy road. Every time you change from a left bend to a right bend, you not only have to turn the wheel the amount you'd expect, but also more to take up the free play.

Even driving in a straight line, the minor corrections you normally make - barely visible with modern steering - would also have to include this free play. Exaggerated steering motion was a normal thing in real life driving.

Combine that habit with the actor's lack of knowledge of what is being projected behind them, and you can see why driving scenes looked the way they did.

In old cars you had to turn the wheel a lot further, because there was no power steering. For a feel of that, push a car when the engine is dead and try to steer it without engine assistance (it's hard!).

Also the actors would have to act driving with no idea what the screen behind them was doing, and often still do.