Four more schools in Birmingham have stopped teaching about LGBT rights following complaints by parents.
Leigh Trust said it was suspending the No Outsiders programme until an agreement with parents was reached.
Earlier this month the city's Parkfield Community School suspended the lessons after protests were held.
Campaigner Amir Ahmed said some Muslims felt "victimised" but an LGBT group leader said No Outsiders helped pupils understand it is OK to be different.
In a letter seen by the BBC, Leigh Trust said it was halting the lessons until after Ramadan, which finishes in June.
The schools involved are Leigh Primary School, Alston Primary School, Marlborough Junior and Infants School and Wyndcliff Primary School.
Leigh Trust - which is yet to comment publicly - said it wanted to discuss the programme with parents to find "a positive way" of teaching about the Equalities Act.
Some parents at Parkfield, and the other four schools, claim the classes are inappropriate for young children and the schools' LGBT message contradicts Islam.
The No Outsiders project was created and piloted at Parkfield in 2014 by assistant head teacher Andrew Moffat, who was made an MBE for services to equality and diversity in education in 2017.
Ofsted has deemed the lessons as "age-appropriate".
Mr Ahmed, one of the leaders of the Parkfield protests, said he had seen a presentation about the programme that was to be shown to the government as part of the school's Prevent strategy - which is aimed at reducing radicalisation.
"I think that's outrageous," he said.
"It's quite disgusting that the school has presented our children as potential radicals."
A spokesperson for Parkfield Community School said: "The powerpoint was written four years ago in line with Prevent duty at that time.
"No Outsiders is all about tolerance, accepting difference and respect, which are all key aspects of community cohesion and our fundamental British values."
Mr Ahmed said his community was "respectful and tolerant" of British values but now felt victimised.
He claimed parents who had protested were "effectively seen as homophobes in the wider community".
"Fundamentally the issue we have with No Outsiders is that it is changing our children's moral position on family values on sexuality and we are a traditional community.
"Morally we do not accept homosexuality as a valid sexual relationship to have. It's not about being homophobic... that's like saying, if you don't believe in Islam, you're Islamophobic."
But Khakan Qureshi, a gay Muslim activist who runs Birmingham South Asian LGBT and was invited to visit Parkfield School last week, said he supported the need for the lessons.
"Myself and many others knew from a young age that we were different and we wish we had this sort of education," he said.
He feels the Muslim community as a whole is not homophobic, but believes a minority within the protesters are "agitating".
"The attitudes of the protesters towards the No Outsiders programme is completely homophobic," he said.
"No matter how they package it, it still comes across as homophobic."
He said given the existing legislation to stop discrimination, "I don't understand why certain communities here in the UK are not adhering to those laws".
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