Is this normal? (in General)

Is it normal for my bows to do more damage in Melee rounds than ranged?

Short answer: Yes

I can't find te wiki page on the Ranged penalties to damage. :'(

I couldn't find a wiki page with information on this either. Yes i understand there are penalties to dex etc. But why is there a buff to my dmg?

The Ranged rounds have built in reduction to damage.

This gets less and less until Melee, where it is zero.

The closer you get to someone, the more damaging you become with ranged weapons. However, you get penalized in melee by only firing every other round. It evens out. Sort of.

Ranged penalties for Magic Damage

Without a Helm of Clearsight 58%/55%/52%/49%/46% reductions in rounds 1/2/3/4/5 respectively
With a Helm of Clearsight 58%/55%/52%/49%/46%/0% reductions in rounds 1/2/3/4/5/6 respectively

Now that is for magic damange, but I'd expect physical damage to be somewhere along those lines as well (maybe not as drastic though)

However in the Wiki ranged penalties for Physical damage are only listed in PTH and CTH, not damage.