Hello, bike forums user, for 2 days I tried to adjust the front derailleur of my mom's new bicycle with my dad. The best guide for this that we found was the Park Tool's video about "How to Adjust the Front Derailleur".
The process was pretty straightforward before we finished adjusting the L-limit screw. For the H-limit screw, the process is to put the biggest cog in the front, and the smallest at the rear. But every time when we tried to do so, the derailleur was shifting over the 2nd cog, that was because of high barrel tension, but every time when we were making the cable slacker, the problem was the same, and turning more resulted in running out of turns. Now, me and my dad thought to make the cable so that it will have more space to go in, and the process was like this: put the smallest cog in the front, and the largest at the rear, disconnect the cable from the derailleur, turn all the way out the barrel adjuster counter-clockwise, and then all the way back in clockwise, and then turning 3(before this it was just 2) revelations counter-clockwise, and then connect the cable back. But when shifting, the problem was the same, and turning clockwise all the way in resulted in the same problem. This was our headache for 2 days! My mom's new bicycle uses the Shimano Tourney TZ. Please, if somebody knows the problem and the fix, please, just please tell it to me. I intuitively thought that the problem might be in shifting levers, and the dysfunction could be that they move the cable too much, and the "pre-determined" levels are not set properly. So please, if you why this problem is occurring, please, tell it to me, I am tired and my dad is tired, I am pissed off, and he is too

A few more details: We managed a few times to put the chain on the 2nd, but when we did that the chain was rubbing pretty heavily with the derailleur chasing, when we tried to slacken off the cable(so that the derailleur will go inward) we ran out of turns, no matter how much space the barrel adjuster had. Plus that when attempting to pull the shifting lever up so that the number on the shifting levers will match the cog number, we were always unable to do it. By turning counter-clockwise the H-limit screw, we managed to pull the chain on the 3rd cog and the shifting levers to show 3, but there was still heavy chain rub, trying to slacken off the cable resulted, AGAIN, in running out of turns, n matter how much space we left reserve.