Ben video izliyorum.

The infinitive is izlemek (to watch). To form izliyorum:
1) Drop -mek → izle-
2) Add the progressive suffix -yor, vowel-harmonized to -iyor → izliyor
3) Add the first-person singular ending -um → izliyorum
Altogether izl-iyor-um = “I am watching.”

-yor marks the present continuous (progressive) aspect (equivalent to English “-ing”).
-um marks first-person singular (“I”).
Together, they form “I am …-ing.”

The progressive suffix -yor begins with a vowel. Since the stem izle- ends in a vowel, Turkish inserts a buffer consonant y to avoid a vowel cluster:
izle + y + or + um → izliyorum.

Turkish uses the accusative suffix (-ı/-i/-u/-ü) only when the direct object is definite/specific. Here video is indefinite (“some video” or “a video”), so it remains unmarked:
Video izliyorum = “I’m watching a video (in general).”
If you mean “the specific video,” you’d say Videoyu izliyorum.

• Use bir for “a”: Bir video izliyorum = “I’m watching a video.”
• Use the accusative for “the”: Videoyu izliyorum = “I’m watching the video.”
  Context alone can also signal indefiniteness: Video izliyorum is understood as “I’m watching (a) video.”

Use the aorist (simple present) with the -r suffix:
Video izlerim = “I watch videos” (habitually).

• Indefinite plural: videolar = videos → Videolar izliyorum (“I’m watching videos” without specifying how many).
• With a number, Turkish usually omits the plural suffix: İki video izliyorum = “I’m watching two videos.”
  If the two videos are specific, add the accusative: İki videoyu izliyorum.