Last week Boris Johnson disbanded an LGBT advisory panel, which had been appointed by Theresa May when she was prime minister, shortly after three members quit accusing the UK Government of dragging its feet on outlawing conversion therapy.
In a speech on Sunday reaffirming a commitment to equal rights and reform of the Gender Recognition Act, Ms Sturgeon said: "And if the UK Government does not take serious action on conversion therapy, an SNP government will bring forward our own legislation to end these discriminatory and harmful practices against LGBT people insofar as the powers of the Scottish Parliament allow."
Mr Johnson has said he would legislate on the issue, but a letter emerged last week in which he suggested churches could "still provide pastoral support (including prayer)... in the exploration of sexual orientation and gender identity", which campaigners say still counts as conversion therapy.
In her speech, Ms Sturgeon said: "Scotland has been on a journey from a nation that once criminalised LGBT people to one that is now recognised as among the best in the world for equality, but we know there is still more we need to do to make Scotland a fairer and more equal country for all."
And she singled out equality for transgender people as an area where more progress was needed.
But the proposed reform of the Gender Recognition Act, in particular allowing trans people to self-identify their gender, has become a hugely divisive issue within the SNP.
Edinburgh South West MP Joanna Cherry – a vocal opponent of gender reform – was sacked from her role as the party's justice and home affairs spokeswoman at Westminster in February.
She has insisted it is not transphobic to advocate for women's sex-based rights, but says she suffered 18 months of "social media lies, smears and foul-mouthed abuse" for her stance.
Ms Sturgeon said in her speech: "The SNP remain committed to improving the lives of trans and non-binary people. Trans people continue to suffer stigma and prejudice and suffer poorer health outcomes relative to the wider population.
"We are committed to tackling transphobia head on through inclusive education and action to tackle prejudice and hate crime.
“In the next parliament, we will work with trans people, women, equality groups, legal and human rights experts to identify the best and most effective way to improve and simplify the process by which a trans person can obtain legal recognition. We remain committed to making necessary changes to the Gender Recognition Act that arise from this work at the earliest opportunity.”
The First Minister said the government would ensure the changes did not affect the rights or protections women currently had under the Equality Act.
“It is important that concerns about GRA reform are addressed through informed and respectful discussion,” she said.
"However, we must never allow them to be a cover for transphobia or disinformation.”