in this pic you see the plane actually turn into the mark

A bit confused...(because I know what I see). The curved contrail, is that your focus? It is exactly as it should look, when a jet making a contrail changes heading (or, makes a turn). Quite normal.

Trying to keep it simple, any airplane in a turn will travel in a curved course, when viewed directly from above or below. This will be defined as an "arc" that, if connected fully, would make a full circle. Like any circle, there is a radius that defines half the diameter. Therefore, an arc (or, just a section of the circle) shares this same radius. Savvy?

Now, the actual radius (of an arc segment, or even a full circle) that is defined by the motion of the aircraft depends on the speed, and angle of bank. Just those two factors. (Let's ignore wind effects, assuming winds are ZERO).

Slower speeds and steeper bank angles result in smaller radii. Higher speeds increase radius, for any given bank angle. Higher Bank angles increase radius for any given speed.

Since commercial passenger jets are limited to 25° of bank (and usually MUCH less, generally not more that 15° at higher cruising altitudes), then really all normal turns have quite large radii.