Why does ionization energy increase as we go from left to right in a period? Why does ionization energy increase as we go from left to right in a period?

In my textbook, the explanation is as follows:

"This is consistent with the idea that electrons added in the same principal quantum level do not completely shield the increasing nuclear charge caused by the added protons. Thus electrons in the same principal quantum level are generally more strongly bound as we move to the right on the periodic table, and there is a generally more increase in ionization energy values as electrons are added to a given principal quantum level."

This, however, doesn't make any sense to me. From what I understand, in a stepwise ionization process, it is always the highest-energy electron (the one bound least tightly) that is removed first. So when we go more to the right in the periodic table, there are more electrons, and thus much more possibility for these to shield the outer electron from attraction to the nucleus. This would make it easier to remove that outer electron, and so I would say the opposite: the ionization energy should lower when we go to the right. 

So can someone explain to me why this is not so? Thanks for any clarifications.
Crash Course on Ionization Energy:


As we all know, atoms prefer a full valence shell. So as we go right in a period, we are increasing $e-$. And also ADDING PROTONS. Because we are adding protons, the size of the atom gets smaller because the nuclear charge will be more powerful. Adding protons in a period trumps the addition of electrons. At the end of the day, we have a small atom with many electrons in it's valence shell that does not want to let go of them. Especially the Noble Gasses and Halogens.
Ionization energy decreases as we move down a group because:

As we move down, a new full energy level is being added. More electrons means more repulsion.
This creates the shielding effect where the addition of the shells, shields the outer electron from receiving the nucleic charge. NOTE: Here, however the addition of another energy level trumps the addition of protons.



This is just a piece of the whole picture.