When the first photon hits the screen and collapses, that doesn’t mean its twin photon collapses too.
Yes, it does. By definition, entangled particles are described by a single wave function. If the wave function collapses, it has to collapse for both of them.
So for example, an entangled pair of electrons can have a superposition of up and down spin before either one is measured. But if you detect the spin of one electron as up, then you immediately know that the spin of the second electron must be down. And if the second electron must be down then it is no longer in superposition, i.e. its wave function has also collapsed.