Learn Farsi: من قضاوتش نکردم، بلکه بخشیدمش. - I did not judge him, but rather I forgave him.

بلکه specifically means "but rather" or "on the contrary." It is used to correct a preceding negative statement. Since the first clause says what you didn't do (قضاوتش نکردم), بلکه introduces what you actually did instead. If you used اما or ولی, it would sound like adding an unrelated contrasting fact rather than replacing a false assumption.

In قضاوتش نکردم, why is the "him" pronoun (ش) attached to the noun قضاوت instead of the verb نکردم?

قضاوت کردن (to judge) is a compound verb made of a noun plus a light verb. In Farsi, attached object pronouns like ش (him/her/it) are normally attached to the non-verbal part of a compound verb rather than the verb itself. So we say قضاوتش نکردم (literally "his judgment I didn't do") rather than قضاوت نکردمش.

Then why does the pronoun attach to the end of the verb in the second half, بخشیدمش?

Unlike the first half of the sentence, بخشیدن (to forgive) is a simple, single-word verb. Because there is no noun part to attach the pronoun to, the object pronoun ش goes at the very end of the conjugated verb.

How exactly does the word بخشیدمش break down?

It is made of three connected parts: the simple past tense of the verb (, meaning "forgave"), the first-person singular ending (, meaning "I"), and the third-person object pronoun (, meaning "him"). Put together, literally means "forgave-I-him."