Learn Turkish: Romanı okuyorum. - I am reading the novel.

The suffix -ı is the accusative case marker. It shows that the novel is the specific, definite object of your reading. Without it, roman okuyorum would mean "I am reading a novel" (or just novel-reading in general), but romanı okuyorum specifically means "I am reading the novel."

Why does roman take -ı instead of another vowel like -i or -u?

The accusative suffix follows 4-way vowel harmony. Because the last vowel in the root word roman is an 'a', the suffix must be -ı. If the word had a different final vowel, like the 'e' in ev (house), it would take -i to become evi.

The present continuous suffix is often taught as -ıyor or -uyor. Why is there no extra vowel before -yor in okuyorum?

The verb root here is oku (to read), which already ends in the vowel 'u'. When a verb root ends in a vowel, it doesn't need an extra connecting vowel before the -yor suffix. So it simply becomes oku + yor + um.

Where is the word for "I" in this sentence?