Per the setuo, what I draw on turn one should cost 1. I have done 2 rounds and with mage and all cards for me and the opposite side were regular cost.
The special rule for this Tavern Brawl does not change card cost. It changes which card is drawn.
On turn 1, it will draw a 1-mana card if you have one. If you have none, you get a “normal” random card draw.
On turn 2, it will try to draw a 2-mana card in the same way. And then a 3-mana card on turn 3, etc.
You misunderstand the rules.
On turn X, you will draw an X-cost card from your deck, if there is one.