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.
High reps do not build high strength. They will help to build muscular endurance (a type of strength), but the Impact on strength output will be minimized.
Let’s take this to a logical extreme. Let’s say you start working out and your maximum reps for pushups at one time is 5. Then, a year down the road it becomes 50. Has your strength increased ten times? No, but between muscle development and the building of your endurance you’ve improved. There will be an increase in strength here, but it won’t be anything grand.
Higher intensity shorter reps grant the most strength gains. Also, try to either work to failure or close to it in order to ensure that you’re actually challenging yourself properly. 12 reps is the standard for a reason, do more or less with an understanding of why. More for endurance, less for strength. Twelve even for a good balance.