I’m a voice teacher based out of New York, so I might be able to be of some help. 
The sound of the voice is created when air comes through the larynx, causing the vocal folds (also called “chords”) to vibrate. There is a correct ratio of how much air is needed to make the folds vibrate correctly; too much causes ‘Pressurization’; the vocal folds can’t handle all of the air, and they actual smack into each other. Often times this pressure coincides with tensing of outer muscles that have no reason at all to be involved with vocalization (neck, jaw, throat, laryngial, etc.). The point is, that’s how most people try to sing louder, using extra air and muscular tension. We lose most of our control that way. Combine all of that with the fact that a huge majority of the people out there breathe incorrectly, expanding their upper chests with air instead of expanding a muscle called the “diaphragm”.
Stand in front of the mirror with your shirt off and breathe deeply. If you see your chest, clavicles, and shoulders move up, you’re breathing incorrectly. Now try this: do the same thing, but stop in mid-breath; see how your laryngial/neck muscles have shortened and tightened, just below your jaw line? That sort of thing robs you of much of the control you’d have otherwise. And the louder you try and sing, the more control you’ll lose. Think of a person with their feet in shackles. Though walking isn’t much of a problem, if they try to run, they’ll fall on their face. The faster they try and move, the less control they’ll have. (-:
With correct breathing and lack of tension, you can access a series of resonance areas in your skull called cavities. The sound goes into these areas and are magnified, as of someone were yelling into a cave. This is the reason many people can open their mouths and have a huge sound come out, they are connected to those areas (as anyone can be). Some people are blessed initially with a natural connection to those areas, most aren’t. Just like anything in life, some people are naturally great at something that other people need to be taught how to do. Doesn’t mean that with training the latter person can’t be as good, if not better, than the former person. The reality is much the opposite, in fact.
If you sound good singing at a speach volume level, than that is VERY good news for you. Matching pitches and having your voice do what you want it to do at that level is one of the hardest thing to learn (or to teach). If you have that inately, you’re off to a good start. Now you just have to learn the physical part.
IMPORTANT TO REMEMBER: singing is an ATHLETIC activity. You’re developing physical excellence by not only strengthening musculature, but also by learning how to use those muscles in conjunction with each other with conservation of motion and correct technique. Just like if you were working with a track coach or a football coach. Technique brings potential ability into reality. Again, just like anything else in life.
I agree with WHITE RABBIT: see if you can find a good teacher in your area. Where are you from? It’s always possible I may be able to recommend someone, I have resources in various states, you might get lucky. Unfortunately, there are some teachers out there who don’t know what the f@#$ they are doing. Main thing that has been SCIENTIFICALLY proven time and time again: tension is a killer. Some people teach that tension is necessary. They are WRONG. And I’m speakig out of proven fact, not opinion.
Wow. I wrote a book here. 
I’ll be happy to help you in any way I can. I’m one of the lucky people who absolutely love what they do for a living, so I can talk about this stuff for hours. (-:
Feel free to email me if you like (that goes for anyone reading this), or simply ask questions on this thread, and I’ll do my best to help you. (-: