In The Turn of the Screw how have gothic conventions been manipulated/adapted to different cultures and times?

Let's answer your question step by step, shall we?
:)

Henry James uses his literary license to refrain from
excessive descriptors, dramatic endings, nor embellished narrative, in his novel,
The Turn of the Screw. Don't get me wrong, the novel IS complex
in its central theme, but it is also written as a typical ghost story to engage an
audience of Gothic readers. This being said, the connection between his style and the
style of Gothic conventions coincide in the
following:

• The use of isolation as way to enable
  the characters to intertwine with metaphysical and supernatural forces.
  


• The treatment of the topic of human psychology to
  explore anger, fear, despair, loneliness, and depression.
  


• The inclusion of death as a subtopic to explain how fate
  is inevitable in human reality.


• The manipulation of the
  atmosphere: Darkness, fog, coldness, dampness...every element that helps the audience
  feel in a very scary place.

These are the
Gothic conventions in the story.

When it comes to cultural
adaptation, it means that people of different backgrounds aim to solve the problem of
the story, at the same time, and within their unique parameters of age, education, and
social upbringing. 

As you can see, there are three
different generations that aim to explain what is going
on: 

The older maid explains how Jessel and Quint
"corrupted", and "contaminated" the children. The young governess claims that the
spirits of Jessel and Quint want claim the children's souls. Then you get the children's
perspective: They do not claim a thing. They cannot see anything. They are simply
oblivious and wish to be left alone.

Hence, Henry James
says everything and nothing at the same time through his use of Gothic
conventions.

He uses the generation gap of his characters
to illustrate how they things they way they see it. Then, he leaves it up to the reader
to make assumptions. Each of the characters explains their situation according to their
personal history and system of values. It is up to the reader to choose which side is
correct.