My question is simple. Is the following sentence correct?
They don't watch TV often.
My English teacher has told me that the only correct option is:
They don't often watch TV.
Is she right?
I think she is wrong. "They don't often watch TV" certainly isn't the only correct option.
As just one piece of evidence you could consider the Google result counts for these phrases (switched to "I" to get more representative data):
Your English teacher could try to make an argument that "they don't often watch TV" would be preferable in writing, or in more formal situations. I am not sure that even this would be true, but it is at least an arguable position.
However, she is just wrong if she says it is the only correct option. Both of your sentences are perfectly acceptable English. They would be understood by any native speaker, and neither one sounds awkward or unusual.
According to this site, an adverb of frequency should be placed before the main verb, in this case watch. So her sentence is correct, in a purist way, but not the only option. It can also be placed at the end, and still be accepted by native english-speaker.
Your teacher is wrong. Frequency adverbs can be placed:
before the main verb (if the verb is not "to be"). Examples: He never eats vegetables. They will always regret doing this.
after the verb (if the verb is "to be"). Example: He is often late for work.
some frequency adverbs (eg usually, normally, often, frequently, sometimes): in the beginning OR in the end of the sentence. Examples: He writes often. We visit them frequently. Sometimes I am late for office.
Some incorrect examples: