How often do typical passenger flights have problems with turbulence?

At the same time, you almost never see turbulence in movies or television..

Of course not. Aircraft scenes are filmed in studios, not in an actual aircraft. Studios sit firmly on the ground, so it takes a lot of extra effort, time and money to shoot a credible turbulence scene, so they will only do this if the story really requires it.

Are all typical passenger flights going to have problems with turbulence?

Yes. I fly a lot, and in my experience some turbulence is quite normal. I'm often aware of it since I tend to work and if it's getting hard to operate the mouse or if I start missing keys a lot, that's an indicator that it's unusually bumpy. Even that is not unusually and happens on every second or third flight.

Your best scale is the state of the crew.

1. Mild turbulence: seat belt sign on, crew out and about
2. Moderate turbulence: crew suspends meal or beverage service but is still active
3. Turbulence: crew is confined to their seats.
4. Stronger turbulence: pilot is on the air making it clear that EVERYONE needs to be buckled down with their posterior firmly planted in the seat and the crew will aggressively yell at anyone trying to get up.

Unless you got up to #4, you haven't seen anything yet. Even that is perfectly safe, I have encountered this frequently with no ill effect to crew, plane or passengers.

you can tell that the plane is not completely under the perfect control of the cockpit

This is a mistaken belief that perfect control equals perfectly flat and level.

Driving down the road, you will frequently drive over a pothole and have the car judder briefly. Driving down a track, you will probably find more frequent shaking of the car. Driving offroad, shaking is virtually guaranteed. None of these situations means that the driver is not in control of the car.

A plane is exactly the same. Turbulence does certainly exist which will cause the pilot to not be in perfect control of the plane, just as road conditions exist which will cause a driver to not be in control of their car. Like a car sliding off the road, you will really know if that happens! Your statement that the plane is "shaking uncontrollably" is simply not correct, any more than the normal road vibrations you get at 70mph indicate your car is "shaking uncontrollably". And if you're honest with yourself, you should recognise that the shaking you're experiencing is not anything different to what you feel on a daily basis in a car.

It's very simple, really. You are not frightened of turbulence. You are simply frightened of being in a situation where you are not in control of the vehicle you're in, and it really is as basic as that. In that situation, you have a natural nervousness about anything like the plane shaking - it isn't something you understand because you're not a pilot, and it isn't something you have any control over. You naturally feel uncomfortable owning a fear of not being in control, so you rationalise this to a fear of turbulence. But in reality the turbulence is just a hook to hang a more basic fear on. I've seen adults have near-tantrums over having to queue for the toilets on a plane, and that equally clearly came down to them being uncomfortable in an unfamiliar environment which they can't control.

My top suggestion is perhaps paradoxical. Get a ride in a small plane or a microlight; maybe get a taster flight in a glider; or get a ride in a slightly larger plane like a Dakota or something at an airshow. In any of those you'll certainly feel a load of lumps and bumps from the air, because they're so much slower - but you'll be in direct sight of the pilot, perhaps within touching distance. You'll appreciate what kind of bumps can happen without the pilot being in the slightest bit out of control. Essentially inoculate yourself against your natural reaction with knowledge and experience. And then when you're in an airliner and there's a bit of a bump, you can think "ah, that's nothing, I know what's going on there".

Turbulence is not a problem. Fear of turbulence is the problem. Turbulence is a normal occurrence. And, the airplane stays under the control of the pilots during turbulence. They are probably more concerned with losing control of their coffee than they are of losing control of the airplane.

Turbulence is a very subjective subject. Some people are more sensitive to it than others. Your perception of it can differ over time and situation. The purely objective rating of turbulence and its effect on the plane and pilot are quite different than the subjective view from most passengers.

The same analogy can be made of boats. You feel the motion of the water more the smaller the boat and less the larger the boat. Yet, someone accustomed to the water (maybe due to occupation) can handle the motion more easily than someone who only occasionally goes out on the water (maybe due to travel). The same is true for cars. Some people are more predisposed to carsickness than others. Even some beginning pilots are subject to airsickness on the smoothest flights until they get their sealegs.

Coming from a background of a frequent flier, current pilot, and former skydiver, turbulence does not bother me. Since decades before becoming a pilot, I would fall asleep on airplanes as soon as the engine came on. The rocking of light to moderate turbulence would lull me into a deeper sleep. The same minor bumps and jolts would give my wife anxiety. Yet, she takes more flights and spends more time in airplanes than I do. Different experiences and predispositions lead to different perceptions.

Two things that I find helpful to those who find it difficult to acclimate to the motion is to have visual reference and control. Whether it is on land, sea, or air, having the abilities to look outside the vehicle, especially at the horizon, allows the body to anchor its perceptions on something other than the physical or vestibular perception of motion. Relying on either the tactile or the vestibular perception of motion alone can result in “illusions” or spatial disorientation. Having that visual anchor can overcome that. Having control of the vehicle yourself as the pilot/driver tends to anchor the feelings of anxiety as well as the feeling of motion.

If you want a better understanding of what I mean by this, take a ride in a small General Aviation aircraft like a Piper Archer. Where I live, summer time is turbulence time even during times of clear and cloudless skies. The only time the air is smooth is at night. Actually, night-time is inherently less safe for reasons other than turbulence, even though it is the smoothest part of the day. During the day, the turbulence increases as the sun bakes the earth. The smoothest part of the day is the mornings. Because of this, I fly with non-pilot passengers only in the mornings and sometimes at night. I fly with other pilots or solo the rest of the day. That does not mean it is any more or less safe at any point during the day. Nor, does it mean I have any more or less control of the plane. I usually fly with a bottle of water or a thermos of coffee. Sometimes, it is just in a cup with a lid. I have only spilled once during even the heaviest of turbulence.

The question strikes me as a bit odd insofar as it's unclear how one should answer it. How often do typical passenger flights have problems. Well, what is a problem?

The next question would be what are turbulences? You feel a bit uneasy? The aircraft shakes a bit? You spill your drink? That's not turbulences (well... it is, just...). Turbulences, that is when the stewardess who is sensible enough to have her hands on the luggage deposit thingie (a puny defense against turbulences, but better than nothing, at least you don't fall on the slightest occasion) tells that jerk who just got up to please sit down, and he answers: "Yeah, I just wanted... PHWATTTT", and they both cling to the ceiling for half a second, then fall down. That is, well, turbulences. It's something I've actually seen once, exactly like this.

Is this "having problems"? Well, for the forementioned two people having undergone the sudden reverse gravity experience, it sure is. That's a seriously bad, hurtful, thing. You wouldn't want to be in their place. For those seated and belted down, or the airplane as such... couldn't care less. Unless this kind of thing happens at take-off when there's only like 30-40 meters to the ground, nobody cares, really.

Airplane under perfect control of the cockpit? Well, what's there to say about that. In fact, this is actually never the case, but it doesn't matter. The forces of nature (wind, among others) are such that a puny little airplane doesn't matter much to them. But in compliance with mass inertia, an airplane generally keeps moving mostly forward most of the time, and it manages to mostly keep its flight level, too. Mostly, that means it can unexpectedly drop a few dozen meters, sure. But as long as there's nothing below (and most of the time there isn't!), that doesn't matter.
Certainly, a sudden gust of wind from behind 10-20 seconds after take-off -- well that means everybody aboard dies. But luckily, this happens very, very, very rarely. So... what to say. You can lock yourself up in your cellar and you'll still find that life is lethal.

I would be worried a lot more about the fact that in modern airplanes, computers have more control over the plane than the actual human sitting in the chair and that both what a pilot must know and be able to do and what he actually can do is being more and more limited.
Sure, it's great when you need less qualification because it means that you can hire cheaper people. But it also means that if you have an incident like e.g. infamous AF447, which was a situation that 10-15 years earlier every pilot would have handled blindfolded and without sweat, it's "trouble". Everyone in the cockpit panicking and shouting, and pulling random levers wasn't precisely helpful for getting out of stall, or for general survival.