Let me help you on this one.
The reason that your range’s current max is about G4, is most probably because you’re not switching gear (literally).
Try to talk in a way when you do when you’re about to yawn, somewhere above your normal speaking pitch. Keep that sound and try to hold it talking lower and lower and lower. Yes, you probably sound a bit like a classical singer (no wonder, because they use a similar formation). This is just practice, though.
Sing a song of your choosing with this formation or do scales. It might sound odd, but again, this is an exercise. Go up to E4 and keep that formation going higher and higher. This formation has some lightness to it, some warmness, but definitely not some yelling sound. Actually, it’s more of a crying sound. You can easily cry in a high pitch, so this will be possible in a singing attitude. If you do this for some time, say several days, you might notice that a singer like James LaBrie is ‘crying’ from E4 and above. Other great theatrical-like sounding singers like Joey Tempest, Dennis DeYoung all have this crying sound, especially in the E4-C5 range.
If you’ve got the hang of it, you could try to get louder, but by being more present, not by yelling. It’s like giving a conference to a huge crowd. Think Steve Jobs. He was very present, but he wasn’t yelling.
That’s it for now.
STORY
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I start with something seemingly unrelated: Newton’s Universal Law of Gravity. We still use it today, because in several cases it works. It’s an easy concept, in many ways, perfectly understandable. However, it’s flawed. The simple reason for that is that the idea of gravity is flawed. Technically, we are not pulled inside the earth from the core of the earth.
Einstein changed all that with his General Theory of Relativity. Due to the phenomenon called spacetime, we are pushed to the earth by an outside force.
Coming back to singing. For a very, very long time, the idea was that one would sing in their true voice / full voice or not.
Then there were methods had the concept of registers. You speak in your chest voice, somewhere starts your head voice, and if your support or vocal configuration is ‘incorrect’ you go into falsetto. Chest Voice, because the resonance comes from the chest, Head Voice, from the head, and falsetto, because it’s wrong (false).
This idea is derived from physical sensation one could get when he sings ‘in chest voice’, ‘in head voice’, etc. There’s already a problem, though. Labeling registers this way gives the idea that the tone is suddenly changed from one sound to another, but there are many singers who seem to have no bridges, no spots where they sounds significantly different. Therefore, the mix-register was introduced (Speech Level Singing concept as far as I know). You could see Mix as the ease and high range of head voiced ‘mixed’ with the power and presence of chest voice. This idea changed the concept of the physical sensations of chest and head voice, because how would that work, both resonating, or both at use? There are quite a number of people who don't have the sensation of chest resonance when in chest voice and especially not head resonance in head voice (I haven’t). For some methods, there's also vocal fry (your vocal folds are not stretched up) and whistle register (super formation of head voice, creating similar acoustics of a flute, resulting in a flute-like sound).
Methods using the register concept this way, let’s call it CHF (though some seem to have adapted/updated to modern research):
Speech Level Singing by Seth Riggs
Singing Success by Brett Manning (former SLS student and teacher) updated to modern research
Raise the Voice by Jaime Vendera
The Rock-N-Roll Singer’s Survival Manual by Mark Baxter
Pro Secrets of Heavy Rock Singing by Bill Martin
The Modern Approach
The late Jo Estill did a lot of research for decades how the voice works inside. What muscles are at use, how they are configured, etc. She created a system—not a method—of the human voice. Some CHF methods had ideas for what is going on inside, but they could only theorize. The idea that the voice is vibrating fully in chest voice and zipping up in head voice was a nice idea, though that’s not what’s really happening (but an effective idea, the succesful method SLS and SS used it all the time).