Seyretmek comes from Arabic, it is the mashed up/squeezed up version of the two words "seyir etmek" İzlemek does not originate from any other language, I believe the root of the verb is "iz" which means mark. They are synonyms and you can use either one of them in many cases. When watching TV, a football game, movie, or tv series you can use both. So "televizyon/film/maç/dizi izlemek" and "televizyon/film/maç/dizi izlemek" are both correct. Here are some cases when you can only use "izlemek": I watched people pass by- Yoldan gelen geçeni izledim. NOT SEYRETTİM Here are some cases when you can only use "seyretmek": We looked at the scenery from the top of the mountain. - Dağın tepesinden manzarayı seyrettik. NOT İZLEDİK I cannot come up with any more examples but "izlemek" is more widely used. "seyretmek" sounds a bit outdated in a way when talking about watching a movies or tv. As a native speaker, I would use "izlemek" in the case of watching a movie, tv series or tv. So I always use "Film/dizi/televizyon izlemek"

They are synonyms for "watch" rooted from different languages. İzlemek (TR) is more common, it generally implies tracking a movement. Seyretmek (AR) is preferred in certain cases such as watching land view / scenery. They can be used interchangeably but sometimes it may seem odd. Seyretmek is also used to imply a continuity in diseases or traffic but these are homonyms of that word.

In the meaning "watching something", they are synonyms. But izlemek has more meanings that seyretmek cannot be used for.

Film, series, theater, etc. "izlenilir", not "seyredilir". Developments are "izlenir", a route is "izlenir" and if someone follows one, it is called "izlenmek". Television is not "izlemek" , it is "seyretmek". The landscape is not "izlemek" , and he is "seyretmek".