You should sing comfortably in ALL YOUR RANGE. Your goal shouldn't only be hitting a G4, but to acquire the technique that allows you to sing comfortably. This means that potentially you could sing higher notes. But that's not because you exercised to increase your vocal range. It's because the technique allows you to be more relaxed and use the correct muscles when you approach certain notes.
Always use your resonance mask when you sing, and don't create artificial tension, which is probably the cause of your wall at F4. Try to search that resonance mask and a "twang" (
) and see if you can imitate a siren as high as you can without going to head voice. If you can reach a G4 doing the siren exercise, you should be able to sing a G4 comfortably.