For this I can increase the number of turn on the primary side N1. Nevertheless I will increase the magnetic excitation H which is equal with some appoximation to H = N1*I1/l. It will then saturated even "higher" my core.
That would be entirely true if the magnetization current remained the same but, it doesn’t...
Increasing the number of primary turns increases the primary magnetization inductance. This, in turn naturally reduces the primary magnetization current because more inductance means a higher reactance at the excitation frequency (presumably 50 or 60 Hz or some other fixed value).
And, if you look at the formula for inductance, you’ll find it’s proportional to turns squared hence, if you were to double the number of turns, you would get 4 times the inductance and, for a given primary voltage and frequency, the magnetization current would reduce by four.
Another example - if turns increase by 10% then inductance increases by 21% and the current that causes saturation reduces by 21%.
Below is the formula for a solenoid but the same applies for a transformer winding (where \$\mu_0\$ is increased by the relative permeability of the core material and \$\ell\$ is the mean length around the core).
So, if you double the turns (for example), the overall effect on ampere turns (and H field) is that it halves. This is because current has quartered but turns have only doubled.
This means that the effects of saturation are reduced.
You can also introduce an air gap to reduce the effective magnetic permeability of the core. This also reduces inductance (by the amount the permeability is reduced) and, this is “corrected” by more turns but, remembering that inductance is proportional to turns squared, there is still a net benefit on reducing saturation.
Suppose my core is actually with a design which make it saturated and
the voltage on the secondary is too high.
The only way you’ll get too high a secondary voltage is because the turns ratio is incorrect. Saturation does not make the secondary voltage increase.
By decreasing N1, I decrease the excitation H1, so the magnetic flux.
Decreasing the primary turns by (say) 2, lowers inductance by 4 and increases magnetization current four times hence ampere turns (and H field) increases by 2 and you get more saturation.