Learn Turkish: Sanat sergisine gitmiyorum çünkü yorgunum. - I am not going to the art exhibition because I am tired.

This is a combination of a noun compound and a direction suffix. First, sanat (art) and sergi (exhibition) form a compound: sanat sergisi. The -si links them. Then, because you are going to it, we add the dative case -e. When adding a case ending to a compound suffix like -si, you must use the buffer letter -n-. This gives us sergi + -si + -n + -e (sergisine).

Is the -mi- in gitmiyorum the verbal noun suffix -me from this lesson?

No, it is the negative suffix! The negative suffix is originally -me, but the present continuous suffix -yor causes the vowel before it to narrow. So, git-me-yor-um transforms into gitmiyorum. Don't confuse it with the conceptual verbal noun -me taught in this lesson; they just share the same letters.

Why doesn't the t in gitmek change to a d in gitmiyorum?

You might remember that gitmek changes to gidiyorum (I am going) because the t sits next to a vowel. But in the negative form, the first thing attached to the root git- is the negative suffix -mi-. Since -m- is a consonant, the t is protected and stays exactly as it is: gitmiyorum.

How does mean 'I am tired'? Where is the verb?