Both sentences are grammatically correct but have slightly different meanings.
I read the news on Twitter that you asked me to = You asked to read the news specifically on Twitter, and then I did.
I read the news that you asked me to on Twitter = You asked me to read a specific piece of news. I found it on Twitter and read it.
In the second, the person did not necessarily direct you to Twitter to read that piece of news. In the first, they did.
Student or Learner