When faced with a mystery such as this, I like to back up and break it down into the basics. So let's start with some basics, and I sincerely apologize if it's so basic as to be insulting. But it's a computer; in the end it all comes down to the simplest of concepts, e.g. 1 + 1 = 2 (or 1 + 1 = 10 if you think in binary).
"Mixing" in a DAW is all about basic arithmetic, specifically addition and multiplication. With four frozen tracks, no active plugins or volume automation and faders at zero, a bounce or export is going to be literally the sum of every sample in the source tracks. Nothing is random, so each bounce or export will produce exactly the same results.
Normally, we route each track to one or more busses, also known as "mix busses" because a mix is a summation and the bus is where the result of said summation ends up. Think of the DAW as an adding machine (if you're old enough to remember adding machines!) where each track is a numerical entry and the bus is the final sum. As long as the numbers remain the same, the grand total will also remain the same - every time you add them up.
Freezing, bouncing, exporting and routing to a mix bus all use the exact same summing process. The only difference is where the sums end up. This is why the mix sounds the same whether listening to the master bus or listening to an export: the math is the same in either case.
So if the math is always the same, how is it possible to get inconsistent sums? Only one way: changing the source data. Going back to the adding machine analogy, if you were an accountant and your sum came up a dollar short of what it should be, you wouldn't assume the adding machine was broken. You'd look back over your inputs and see which one had been incorrectly entered. Similarly, if a bounce sounds different from an export, then it couldn't have been the same data that got summed.
First rule: always use a master bus. Second rule: make sure everything goes through the master bus. Everything. Verify that by muting the master bus - everything should go absolutely quiet. The bus in turn should always be routed directly to your audio interface. IOW,
the only way anything gets to your audio interface is through the master bus.
Now you have a dependable reference. The sound you hear during playback is the sum of every track routed to the master bus. That's exactly the sound you should hear when you export the mix or bounce the entire mix to a track. If an export or bounce ever sounds different from that reference, then you have made a mistake.
How many ways are there to make such a mistake? Too many to enumerate, unfortunately. Just when I think I've found them all, a new one arises to bite me in the arse. But the best preventative is to establish a procedure and then consistently stick to it.
For starters, I'd suggest that bouncing multiple tracks to a new track is almost never needed in the digital world. If you want to treat 4 tracks as a single sound source, route them all to a bus instead (which in turn is routed to the master). Then you'll be able to adjust their collective volume with a single fader and to apply a single set of effects to all of them as a group. If you want to combine them into a file, perhaps to import into another program or project, then just solo the desired tracks and do a normal export.
As long as it goes through the master bus, it will sound exactly the same as when you were previewing it.
I'd also suggest that normalization is also rarely needed or desirable. For setting the ultimate perceived volume, use a limiter on the master bus. It's perfectly acceptable for a raw mix to have an average RMS of -20dBFS. A separate step called mastering is used to raise the final level to the desired target. Mastering is quite different from the mixing process, and in the professional world is a separate, specialized discipline all its own. For us DIY mortals, it's basically an EQ, a compressor and a limiter on the master bus and lots of trial-and-error until it sounds good.
Sorry if this was overly basic, but it's really all about the basics. Also sorry about the rambling stream-of-consciousness brain-dump. It's still early over here on the other side of the world.