Normalizing steel is a term generally used in reference to processes that heat steel to a high temperature (if in open air with simple steels) and then allow cooling without much of a hold time, or more commonly with electric furnaces taking over discretion from just about everyone, held at a prescribed temperature or set of temperatures in an electric furnace.
During normalizing, the high temperature or more commonly - the controlled soak - is resetting steel grain thoroughly and bringing the alloy to a certain point with alloying elements in solution. For steel with iron carbides (like japanese steel), this will mean more of the carbon is dissolved from the carbides and distributed back in the steel matrix. If you only take away one thing from this whole idea of normalizing, imagine it being a reset that you can always return to. Forging or perhaps other treatments will leave the steel in a condition that isn't ideal. Those other treatments may just be manipulating the steel to get alloying elements in fewer and bigger carbides to make it as soft as possible so it will machine easily.
Heavy forging tends to distribute carbides in less than flattering ways, and presumably, it could also lead to a lot of alloying elements that should be in carbides being melted back out into the matrix where they could lead to higher hardness but brittleness, or some other differences. If you imagine a cracked earth picture -like barren land in a drought - with tennis balls stuck in the cracks, that's a reasonable depiction of grain boundaries with carbides residing at grain boundaries. We'd prefer carbides to be spheres, and in some cases, elongated tubular shapes are OK - something that happens in rolled material where the carbides stretch in the rolling direction. There are treatments that will result in carbides becoming more spherical, but that's a separate topic.
After forging, however, carbides can form along grain boundaries looking almost like a grout between grains. The result of that is lower toughness.
Normalizing will resolve that, or at least a great deal of it.
If you are operating without a furnace, as I am, after forging, I drive steel up to forging heat two or three times and then allow the steel to air cool. Forging heat is a lot higher than the soak that would be prescribed, but we are working in the open atmosphere and if we're going to have steel at heat for less time, we need to make up for it by increasing the heat level.
What you don't want to do as a person using discretion is try to imitate heat treat schedules that would be done in an atmosphere controlled furnace. It's a pointless exercise - controlled temperature normalization soaks can be long and the temperature matters a lot when duration is long.
Decarb (carbon migrating out into the atmosphere) can obviously be a problem at forging temperatures, but it takes duration. We are not going to provide any kind of duration to high heat when heat treating, normalizing and so forth, and thus should not see anything signs of decarb in the finished product. if you do very large amounts of forging and what you're making sees heat as an entire item over and over, you could very well see decarb. if you're reading this because you're a beginner, forging is only something to venture into after you've got really pristine heat treated work from rolled or round stock with material removal.
We don't quench something that's heated for normalizing - but rather let it air cool. If you want to use a steel that's air cooling and do a lot of forging, I'd advise against it, but you can ignore my advice if you'd like. the more hardenable steel is with additional alloying elements to allow fully hardening while cooling at a slower rate, the harder it's going to be to do something meaningful in the open atmosphere. A2, for example, has no normalization process and the furnace heat and slow cool to anneal the steel to make it workable, presumably standing in place of normalization, prescribes a 20 degrees F per hour decline in temperature (think about that, one degree per 3 minutes) decline. You can probably figure out some way to get a good result that has nothing to do with the schedule, but what you'd gain by doing that vs. just using another steel, I don't know what it would be.
if you are dealing with bar stock that is in a good condition to just be heated and quenched, you can ignore this. Simple steels like 1084 really aren't going to vary much, and others like O1 or 52100 if you get as far as trying to master 52100, will have different results with coarse spheroidized. 52100 can be drastic if you try to harden from coarse spheroidized, varying using standard heat cycles in a furnace so far that you could see anything from 57 to 68 hardness. Larrin Thomas did an interesting experiment actually showing that. AT 68 out of a quench, 52100 has a the potential to be a great chisel. at 65, it will temper back to 60/61 and be problematic, and at 57, I guess you could just use it as quenched for a machete - it would make a terrible tool. if you're a beginner and trying your first heat treatment, find something fine spheroidized or annealed, and not coarse spheroidized. Call the supplier and ask if you can't figure out what they're offering.