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
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.
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
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).
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)