When I take commercial flights, or you see turns in the vapour trails, they turn very sharply. What's the reason for this - wouldn't it be more efficient to turn gradually?
Airplanes are expected by air traffic control to turn at a "standard rate of turn" which is 3 degrees per second. This will provide a 90 degree turn in 30 seconds. A standard rate is used so that controllers can anticipate how long it takes for an airplane to turn to a heading and they can line them up easier. It also makes the turn happen in a smaller amount of airspace so that two airplanes can be flying in the same general vicinity without getting too close.
Unfortunately, at the speeds that jet airplanes travel this would mean a very high angle of bank. A standard rate turn at 200 KT TAS is about 30 degrees of bank and jets can go much faster than this, which would require even higher bank angles. They actually limit it to 25 degrees of bank when using a flight director, mainly for passenger comfort.
It can be impressive when looking out the window and seeing the ground, but don't worry, the airplane is more than designed for it!
This is used mainly at lower altitudes where there are lots of airplanes. Once they are up high and airplanes are further apart (and the turns are also usually smaller) they can bank even less (by using a "half-bank" mode) which adds to passenger comfort.
Airliners actually turn relatively slowly, but are often so far away (when viewed from the ground) that they appear to be turning sharply. An airliner cruising at 35,000 feet is actually over 7 miles away, which makes scale hard to comprehend.
From the air, riding in an airliner, you may be confusing rate of turn with roll rate or angle of bank. An airliner will roll relatively smoothly into a bank angle of up to 30° (which can seem steep), and a rate of turn (change in compass heading) of between 3° and 1.5° per second. You often don't feel the roll, but will observe the bank by looking out the window.
The reason for the 1.5-3° turn rate is so that Air Traffic Control can have some way to predict where an aircraft will be in x seconds. Rate of turn is standardized, making that possible.
Most aircraft tend to make standard rate two-minute turns, as in it would take them two minutes to turn 360 degrees (3 degrees per second). At higher altitudes and higher speeds (or heavier aircraft) they might do half-rate (four minute) turns, which are 1.5 degrees per second.
So at a higher speed (which high-flying planes are likely to be at), to turn a the same rate, they must bank more sharply. That's why at higher speeds the halve the rate, so they can reduce the required bank angle.
More likely though, it's just your perspective. From your position on the ground it looks like they're turning sharply but in reality it took them several miles to make the turn. From your position in the plane, it's difficult to comprehend how fast you're going.
Finally, at higher airspeed speed sharper bank angles are required to keep the plane coordinated during a standard rate turn. This means that when you turn, you feel like you're being pressed down into your seat instead of sideways out of it (like when you take a turn fast in a car).