The reason is actually to help them aim faster by utilizing a technique known as "Flash Sight Picture."
I wouldn't advise it though. I am a former Marine Corps marksmanship instructor. I specialized in pistols and have fired these weapons thousands of times. That said, the thought has crossed my mind. The answer didn't come to me until another coach (from the "hood") gave me a good reason why this technique would be used. In practice, it actually does utilize one very important sighting practice, but fails overall. This is a good idea in theory, but fails miserably in the actual execution.
First, a primer on aiming a pistol. (Oh crap, now I have to do a legal disclaimer. See bottom.)
Sight Alignment: Sight alignment is how you line up the weapon to aim. You need to remember that the trajectory of the round doesn't automatically point at the target. (Why I hate most movies with shooting in them.) You have to have control to make sure that both the back of the weapon and front are pointed at the target in a straight line. The easiest way to do this is what we were taught as "building the castle".
You build the castle by lining up the three "turrets" or posts into a formation where the tops are all even and the posts all have even spacing between them. This diagram shows what having an improperly built "castle" or rather how, having improper sight alignment will cause your sights to go off. (The bottom is the correct method, however I disagree a bit on the sight picture element, because I believe in aiming center mast, as I have shown below. Besides this, as long as you aim in the same place every time, you will still be all right.)
Sight picture: Sight picture is when you place your perfectly aligned sights, or in this case, the same thing every time, over your intended target in the same place. The normal practice is to place the sighting posts over the center of your target. This creates a picture of how you should aim. With weapons that don't allow you to change your sights, you need to figure out how to aim and always offset the weapon to where the bullet will hit where you intend it to instead of where you are aiming.
As long as you know how to offset your weapon, this is not a problem that you are aiming low and left if your weapon shoots high and right. So technically, if you just aim the right the same way every time, you can predict where the bullet will go. This is why sight picture is so important. You know where the weapon is in relation to the target. Below is what a good sight picture with good sight alignment looks like.
Flash Sight Picture: This is a technique the military and other organizations use to get on target fast. Really fast. It basically means that that picture you created when you took your time is now placed extremely fast over the target and fire. You don't get exactly the right spacing between your sighting posts, and you don't get that picture in exactly the right place, but you get it generally in the right place. The important thing is that you're close enough to the target and that you have enough experience to hit it without having perfect aligned and centered sights. This takes a great amount of time and use with the weapon before you will be any good with flash sight picture. This is a very quick way of aiming decently under time intensive periods. This is much more common in combat style shooting than the marksmanship style methods of better shooting.
So that is why Gangsters aim the way they do. Confused? You should be. It is a good way to get a flash sight picture, but absolutely horrible for getting sight alignment. Let's take a look at why. The red arrow is the focal point for sideways shooting. This is what will be used along the side of the weapon to get the sight picture.
The correct way should look much more like this.
So you see, there is a rational method to the way that "gangsters" aim their weapons. It isn't an extraordinarily good one, but it does exist. What it is lacking is the ability to aim well and to fire the same shot twice. This is precision, and tilt style lacks it. What it does have is speed, and if you're lucky, speed can be a good asset. However, the Marines don't think this way. It doesn't really matter how fast you can aim if you can't hit what you're aiming at.
Legal and Idiot Disclaimer: Don't take my post as actual instruction on using weapons. I am an expert because I have excellent training, years of practice, and hands on experience. You don't have that, so you aren't an expert. Don't try this at home. Don't practice using these techniques unless in safe conditions with supervision. Don't use guns to hurt people. Hell, don't use them all. Run if you see one. Call the cops. Do whatever, but it wasn't my fault you got yourself shot, and I can't pay your medical bills anyway.