Watching an American TV drama

This should have an article unless the topic is watching American TV dramas in general (in which case it should be plural)

Last weekend, iI watched a new American TV drama, "All hHer fFault".

Titles should be capitalised. You can also place it in quotation marks to make it extra clear.

The drama is about a rich family that thierwhose child was kidnapped.

It is a hotcompelling drama, the story is very attractiveinteresting.

I'm not entirely sure what you wanted to say here but the original sentence gave an impression of the drama featuring sexy scenes and attractive actors, which I'm not sure you wanted to say.

In the drama, a rich family employed a babysitter who was only 20 or 21 yeas old.

If this is the same rich family whose child was kidnapped it would be better to write "the family" instead of "a rich family". The sentence as it is gives the impression that it's a different rich family

She ihas no any experrience of being a mom.

I don't know if it can bethat could happened in reality.

Here you could replace "could happen in reality" with "is realistic".

Because no one believes that the girl has the ability tof babysitter.

Anyway, i never wouldI would never like such a young woman to take care of my child.