Transformer: Number of turns om primary winding
I guess it must be the relation between number of turns and current
  that gives the answer


Correct - amps x turns is called magneto motive force and it's what drives magnetism and potential saturation. It's related to the magnetic field strength (H) by dividing MMF by the distance the flux path takes around the core so: -

$$\dfrac{MMF}{\ell_e} = H$$

And H is related to flux density (B) by the permeability of the core material: -

$$B = \mu\cdot H$$

So, too much flux density goes right back to having too many ampere turns.

For AC, the inductance of a transformer winding is proportional to turns squared so, if you double the turns you get 4 times more inductance and one-quarter the current for a given voltage/frequency. This generally means that more turns means less core saturation because although the turns might (say) double, the amps reduce to a quarter.

The picture below (hopefully and intuitively) explains how inductance is proportional to turns squared: -




  Anyone with a good advice?


That's dependent on the reader!