Turner, who plays Sansa Stark in the hit series, posted on Twitter to say that people with a platform can use their fame to "help get rid of the stigma".
It came after Good Morning Britain host Morgan quoted an article on the social media site, which featured Coronation Street actress Beverley Callard questioning stars' motives for discussing illnesses such as depression and anxiety.
Callard, who plays Liz McDonald, said there was a danger it could become "fashionable" to talk about mental health.
Morgan said: "She'll be hammered for saying this, but it's 100% true."
In reply, Turner wrote: "Or maybe they (celebrities) have a platform to speak out about it and help get rid of the stigma of mental illness which affects 1 in 4 people in UK per year. But please go ahead and shun them back into silence."
The 22-year-old then added: "T***."
The actress later shared several tweets about mental health, criticising those who make light of the subject.
Responding to a comment from one Twitter user, Morgan said he believed that "those suffering from genuine mental illness need proper help & support. Those just shirking responsibility for normal life challenges need to toughen up. We should teach mental strength & resilience in schools. This concept is only offensive to snowflakes."
Former Great British Bake Off winner Nadiya Hussain was among those to wade into the argument on social media, saying "you haven't got a clue" in response to Morgan's initial tweet.
Approximately one in four people in the UK will experience a mental health problem each year, according to the charity Mind.
Turner has spoken out to raise awareness of the importance of mental health before, saying in a message to her followers on World Suicide Prevention Day earlier this year: "You're stronger than you think. You will get through this. You're so loved. You are NOT a burden."
Last year, Morgan came under fire for tweeting "Man up, Britain" in response to reports about mental health statistics.