GFCI not working with shared wire
So did you wire the GFCI "the same way" which would mean ignoring the warning tape and connecting (probably) hot to the LINE hot and neutral to the LOAD neutral? [less probably the Hot to LOAD hot and the neutral to LINE neutral, given the "usual" interpretation of "top")

That won't work and would tend to explain the problem.

Try connecting both wires you have to the LINE terminals, ONLY, and see what happens. Ah, your pictures indicate that you have more wires than you mentioned - still, the problem is almost certainly due to the fact that the two screws on a "normal" outlet are tied together (unless the "tab" is broken, usually on the hot side only, which is very common in kitchen outlets wired as a MWBC - but usually that brings a red wire into the picture.) The screws on a GFCI are different - one set is the LINE where power comes in, the other set is the LOAD and normally under protective tape in hopes that the person installing it will realize that they are different than normal outlets.

You can wire-nut all the whites you have together with a short white (commonly known as a pig-tail) and connect THAT to LINE neutral, and wire-nut all the blacks together with a short black and connect that to the LINE hot, which should work. 

It will likely also work (and GFCI protect something else, but we don't know what yet) if you put one "set" of wires on the LINE and the other "set" on the load, rather than "crossing" sets as you have it - i.e. probably the "looped" wires are a set, and the end-stripped wires are a set, and one or the other of those is also the connection to power - I'd guess the looped set, but that's just a guess. If it does not work that way, swap sets and try again.