Learn Turkish: İyi ki o adama bulaşmadın. - Fortunately you didn't get involved with that man.

İyi ki means "fortunately" or "it's a good thing that." It acts as an evaluative declarative marker and simply sits at the beginning of the sentence. It doesn't force any special grammatical changes; you just follow it with a normal, fully conjugated verb like bulaşmadın (you didn't get involved).

The English translation says "with that man." Why does Turkish use the dative case (adama) instead of the word "with" (adamla)?

The verb bulaşmak requires the dative case (-e/-a) to show who or what you are getting entangled with. While English uses "with", Turkish conceptualizes this action as directing yourself toward or onto the person or source of trouble.

What is the exact feeling or nuance of the verb bulaşmak in this context?

It has a strong negative nuance, meaning "to mess with" or "to get entangled with" someone who is likely trouble. Literally, bulaşmak is used for things that smear, stick, or contaminate (like dirt, wet paint, or a virus spreading to someone).

What are the suffixes doing in the word bulaşmadın?

It breaks down as bulaş-ma-dı-n. bulaş- is the verb root, -ma makes it negative, -dı is the definite past tense, and -n is the "you" (singular) ending. Altogether it means "you did not get involved."