It is believed that high repetitions (more than 12) build muscle endurance and not strength, but to me this doesn't sound right, here's why, let's suppose for example that i can curl 20 pounds for 5 reps, then i choose to progress by increasing my reps instead of weight, therefore after some time i could curl 20 pounds for, say, 15 reps. This means that i turned my 5 rep max into my 15 rep max, so now my current 5 rep max had definitely increased, hence i built strength by using high reps. What's wrong with this reasoning? Is it the case that high reps build strength as well as low ones?
I think you've heard the right idea, but you're misinterpreting it.
Of course high reps builds some strength, but certainly not a whole lot. It's going to take a lot more effort (in terms of joules spent) to increase your 5RM by doing 15-rep sets, than it would to increase it by actually doing 5-rep sets.
Don't think of it as "it does, or it doesn't". Most activities in the gym build strength. Just not to the same degree.
It's not a dichotomy. It's a spectrum.
Imagine a sort of sliding scale where muscular endurance is on the high rep side and strength output is on the low rep side. The middle area will be approximately 8-12 reps and deviations from there start to focus more on one aspect than the other. Both aspects exist regardless, but the one aspect will start to overshadow the other the further you go from the middle.
Pushups are an interesting example of this, if someone can only do 5 reps to begin with they will primarily be building strength at first. But pushups are easy to progress with and after a year that same individual should be able to do 20+ reps. At that point any strength gains are almost negligible.
If strength is a priority, focus on doing sets in the 5-8 rep zone. If muscular endurance is a priority, focus on doing sets in the 12-20+ rep zone. If you have no preference, working on various rep zones will help you to challenge yourself in different ways. When training, you should always perform your sets to or close to failure in order to ensure that you provide your body a good stimulus for improvement. What is failure? There are two types, but here I am referring to technical failure where you can no longer perform another rep in good form. In decreasing order of importance, progress in the gym is hinged on the following: consistency, quality (of technique), Intensity, and strategy (having a plan). Rep ranges mean almost nothing if you fail to provide sufficient intensity by never coming close to failure. You can certainly build strength at a higher rep range, but it is in a lesser capacity by comparison.
It's about the nervous system: Maximum strength is determined mostly by, among other things, the number of nerves activated during the motion. Training with higher weight in the low rep range increases the number of nerves activated over time. Your brain actually adapts to the training by firing more neurons. More neurons means more individual muscle fibers will be activated. More fibers means more maximum weight lifted.