I would like to understand what is the difference between the plate of a capacitor and an infinite plane sheet of a Charge? I mean, when we use Gauss' law to find the electric field between to plate of a capacitor, we find $\rho/\epsilon_0$ (if no dielectric is inserted), with $\rho$ the surface charge density of the plate, and we find $\rho/2\epsilon_0$ for a infinite plane of a surface charge density $\rho$.
If we consider the field from an infinite plane with charge density $\sigma$ then as you say we get a field looking like this:
Now suppose we put a second infinite plate (shown in red) with a charge density $-\sigma$ below the first one:
When we have a system of charges the total field is just the sum of the individual fields, so to get the total field from the pair of plates we add the two fields together and this gives:
And we end up with a field $E = \sigma/\epsilon$ in between the plates and no field outside.