You, if you have non-African roots, have 1-4% Neanderthal DNA. Roughly, we can say that means you can take a slice of unrelated ancestors way back that's 1-4% Neanderthals. Each of their kids had 50% of the Neanderthal genome, and, assuming the next incoming ancestor was fully Homo Sapiens, had grandkids with only 25% of the neanderthal genome.

Since there's a lot of people and a lot of interbreeding events, you'd naively expect it to be a completely different 50% every time, and collectively contain most or all of the whole thing. However, not every Neanderthal allele is equally likely to be passed down, so that's not actually what happens.

I don't know how much of it actually remains across the human population exactly, but I do know parts of the X chromosome are complete deserts of Neanderthal DNA, at the very least. Like I went into elsewhere in the thread, that's a pattern that indicates having Neanderthal admixture there causes sterility, and so male offspring with only one copy of the chromosome don't reproduce, and don't appear as an ancestor of yours. Those segments of the Neanderthal X chromosome are gone in living populations.

Edit: Reading what you wrote again, I think the detail you might be missing is just that lots of people die with no descendants, and the carrying capacity of ice-age Europe was finite. It's not like the two lines just fused together without a change in size; the mostly-human population slowly grew and the mostly-Neanderthal population slowly shrunk over a few millennia.

It's not known why, or how exactly that went down. It could be a reproductive quirk, or just humans being slightly better somehow. It's probably wasn't organised genocide, though, for quite a number of reasons.