LTspice - Flyback simulation misunderstanding
What I do not understand is why I(D3), the output diode of Vout1 take
all the current from the primary when the voltage at the secondaries
(across L4 and L2) is superior to Vout1 and Vout2. If it is the case
the two output diodes should conduct as the voltage across the diode
are forward bias.

The main reason is that the load current on Vout1 is twice that on Vout2 but you also need to consider that the diodes are not ideal in the real world.
So, you are not thinking about diodes in enough detail. If you had a variable power supply, a voltmeter and an ammeter and you plotted the forward voltage of a diode against the current it takes you would find this approximate rule: -
$$\boxed{\text{A 100 mV reduction in the applied forward voltage reduces the current ten times}}$$
In a comment you say this: -

The both should conduct if the voltage across the diode is foward
biased whatever if one output voltage is lower than the other ? The
difference between the two outputs is about 50 mV

A 50 mV difference in the forward bias voltage of two identical diodes will result in about a 3:1 difference in forward current. Look at the data sheet for ON semiconductor's 1N4148: -

Picture from this answer.
Concentrate on the left diagram - at 0.5 volts applied, the current is 100 uA. Now drop down 100 mV to an applied forward voltage of 0.45 volts and can you see that the current has dropped to about 33 uA?
That's a 50 mV drop causing a 3:1 reduction in current.