NOT A TEACHER
"on the outskirts (of something)" is idiomatic. See here. ". . . . a new house on the suburbs of the city" doesn't work.
Student or Learner
What is the difference between suburbs and outskirts?
In the following senctence you can choose between suburbs and outskirts and outskirts is marked as the right answer:
My uncle and his family have just moved into a new house on the outskirts of the city.
A suburb need not be on the outskirts of the city. Some suburbs only have other suburbs as neighbors.
Also, you would be "on" the outskirts, but "in" the suburbs.