Keeping in mind, the usual disclaimer about not being a doctor; my medical knowledge being limited, and I’m going to be simplifying some of this a lot?
In general? No. Gunshot victims usually die from blood loss. This can happen pretty quickly, or it can take awhile, it all depends on what the bullet damages. But, it’s not usually an instantaneous process.
I could swear we reblogged an infographic on blood loss symptoms by volume, but all I can find right now is this.
If a bullet nicks or severs an artery, this will result in rapid blood loss and death. We’re still talking about at least a couple minutes before death. With non-vital areas, where there’s only soft tissue damage, you could be talking about bleeding out for hours.
If the bullet damages internal organs, then things get a little messier. If the bullet manages to destroy something you actually need, like the brain or heart, then you’re going to die.
That’s the simple part… here’s the part that’s slightly outside my expertise. Hydrostatic shock is where a gunshot strikes someone, and kinetic force from the gunshot is transmitted through the fluid in their body to injure other organs. I’ve seen the discussion go back and forth over the years, some people insist that hydrostatic shock can rupture internal organs and cause people to suffer lethal injuries from relatively minor wounds, and there is some support to the idea that hydrostatic shock result in minor hemorrhaging. The other side of the argument is that, while the shock wave does occur, it isn’t actually life threatening. (There’s also a contingent who’ve been waging an edit war on the wiki page claiming that hydrostatic shock doesn’t exist at all.)
The actual hydrostatic shock wave increases the more a bullet disrupts the tissue. So .45 hollow points will produce more shock than a jacketed 9mm. These can result in minor brain hemorrhaging, but this usually comes up in autopsies, meaning they probably died from getting shot, and it turns out there was additional trauma that went undiagnosed.
If I’m sounding like I don’t know if this is relevant or not, it’s because I don’t. Ignoring specialized hydrostatic shock rounds (again, like hollow points, high explosive rounds, or anti-materiel rifles), I’m not sure if hydrostatic shock actually kills people, or just slaps an extra layer of damage on an already life threatening injury.
Bullets in bodies are extremely unpredictable, but when you’re writing, a basic rule of thumb is “no, gunshot wounds are not immediately fatal.” In fact, if you discount suicides, only about 30% of all (intentional) shots are even fatal.
The most likely outcome from getting shot during a fight is that the person will keep fighting. Even if a wound is eventually fatal, a person will still be conscious and operational unless they’ve been hit in an artery or vital organ, so they can still fire their own gun at you while bleeding out. Getting shot sometimes feels a lot like getting punched (if there’s only soft tissue damage), and some people can run around for several minutes or even hours without realizing they have been shot. Even shots to the torso are survivable. You can run around for a surprisingly long time with a hole in your lung or shrapnel in your liver. (You’ll pay for it later, but since we’re talking about ‘immediately after being shot’…)
The two biggest concerns with gunshot wounds are bleeding and shock. (Not the hydrostatic shock — I’d actually never heard about that — but hypovolemic shock.) Shock can be fatal even with minor wounds.
The main thing to remember when writing gunshot wounds is that bullets in bodies are unpredictable. Someone can live through being shot 20 times and then die from being shot once in the leg. You can (almost) do anything. If you want to strive for realism, it’s all going to be dependent on the characters’ reactions to events. If a character is familiar with guns and shooting, they should expect a body to keep moving after being shot, and if they’re trying to stop they person they should shoot multiple times. “On shot, one kill” is a brag, not a methodology. If a character is unfamiliar with gunshot wounds, they might (like you) be surprised to find you can’t actually make someone fall over with one shot. And, of course, if you’re going to go the route of “Whitley said I could kill someone by shooting them in the shin” or some other possible-but-unlikely scenario, every character involve should be boggled so that the audience knows this is an unusual thing.
But, yeah, like 90% of the time, someone who gets shot is not going to die instantly.