My illuminated friend, there are a couple of drills you can try to prove to yourself that your handgun is doing its job, and to show you how and where you need to improve.
First drill: set up a target (doesn't really matter what kind, but one with a clearly-visible bulls-eye is probably more useful for this drill) at 25 yards in front of a shooting bench (or a steady, firm table if a purpose-built bench isn't available). Sit down at the bench, and use sandbags or some other firm support to build yourself a shooting platform so that the gun can be held in line with your eye and the target without too much contortionism!
Hold the gun in your shooting hand, and support that hand with the other one, in such a way that the support hand is resting on the sandbags, but the gun itself is not touching anything. Make sure that your shooting arm is fully extended - don't break the wrist or the elbow out of a straight line between your shoulder and the gun.
Fire a group of 5 rounds, not trying for speed, but shooting for the maximum possible accuracy. Your handgun should be capable of shooting a 3"-4" group at 25 yards without any trouble. If the group is larger than this, slow down and fire another one, using a different point of aim, and being really slow and deliberate, especially with trigger control. Your objective is group size, not bullseye hits - just use exactly the same point of aim for every shot, never mind where the bullets are actually hitting.
Once you've finished, check your group size. If it's in the 3"-4" range, you're doing fine: if it's larger, your shooting needs work. Then check the group location (i.e. the center point of the group) in reference to your point of aim. Don't worry about elevation at this time: but correct your sights so that the windage is allowed for (e.g. if your group is 4" to the left and 6" below the point of aim, drift the rear sight to center the group, but don't worry about elevation at this point). Fire another test group in the same way, and get the sights properly adjusted.
If you get the group size right, and the windage properly adjusted, you're more than halfway home. The rest is practice, from the bench and other positions, and getting the sight picture and trigger control right.