Many lesbian, gay and bisexual people continue to live in fear of prejudice, despite the significant progress in recent years to improve equality.

Gay in Britain, a report by the campaign group Stonewall, suggests that large numbers of Britain's 3.7 million gay people fear they will be discriminated against if they stand for political office, seek to foster a child or look to become a magistrate.

Three in five of those questioned in the YouGov poll of just over 2,000 gay people said they expected their child would be bullied in primary school if it were known they had gay parents; more than four in five expected the same if they had a child in secondary school. Seven in 10 people expected to face barriers because of their sexual orientation if they applied to become a school governor.

Despite the move to legalise gay marriage, the survey also suggests the Conservative party still has some way to go in winning the support of gay, bisexual and lesbian people. Three-quarters think they would face barriers from the Tories if they wanted to stand as an MP, while more than a third would expect to face barriers from the Labour party and more than a quarter from the Lib Dems.

Both the coalition government and recent Labour administrations have done much to advance the cause of gay equality, but it appears many gay people fear societal attitudes lag behind parliament.

"Completion of our work on marriage means that one strand of Stonewall's domestic focus – legislative equality – is effectively complete," said Stonewall chief executive Ben Summerskill. "But this polling demonstrates starkly that changing laws doesn't change attitudes and lives overnight."

Discrimination hit the headlines again last week when a driver with the London minicab firm Addison Lee was suspended while he was investigated for allegedly calling a gay couple "dirty" and kicking them out of his cab after they held hands.

The report found that one in five lesbian, gay and bisexual employees have experienced verbal bullying from colleagues, customers or service users in the past five years. A quarter said they were "not at all open" to colleagues about their sexual orientation.

It warns that the criminal justice system presents barriers for lesbian, gay and bisexual people who fear they will be treated worse than their heterosexual counterparts, whether they are reporting crimes, suspected of committing crimes or dealing with police and prison staff. Almost six in 10 gay and bisexual men and almost half of lesbian and bisexual women believe they would be treated worse by a prison officer than a heterosexual person.

Eight in 10 lesbian, gay and bisexual people would expect to face barriers if they applied to become foster parents. Almost half expected to be treated worse than a heterosexual person by an adoption agency if they want to adopt a child.

"Gay taxpayers contribute £40bn a year to the cost of Britain's public services," Summerskill said. "They should be able to have confidence that they will receive the services they need when accessing schools, hospitals or policing. It's time the needs of this country's 3.7 million gay people, both as citizens and service-users, were properly met."

Stonewall said its report showed that, in contrast to claims made by anti-gay campaigners, there is overwhelming support among gay people for equal marriage. Nine in 10 lesbian, gay and bisexual people support the government's recent moves to extend civil marriage to same-sex couples.

Significantly, two culturally important areas of British society appear to be of concern to gay people. Stonewall said sport, with its significant place in British life, had failed to create a welcoming environment for gay people.

Its survey found more than six in 10 gay and bisexual men and four in 10 lesbians and bisexual women expect to experience homophobia if they take part in a team sport and are open about their sexuality. And almost half think that television's portrayal of lesbian, gay and bisexual people is unrealistic.