Minister for women and equalities Penny Mordaunt announced the money alongside plans to launch an LGBT Advisory Panel to advise the government on policy, act as a sounding board, and provide evidence on LGBT people's experiences.
In schools, voluntary and charitable organisations are being offered a share of up to £1m to extend an existing anti-bullying scheme that is currently operating in 1,200 schools. The money will also be available to provide teacher training on how to spot early signs of bullying, how to intervene, and to provide teaching resources on LGBT issues.
Meanwhile, a further £1m will be available to organisations focusing on young people's health and social care to address concerns over LGBT people reporting negative experiences while trying to access services.
The remaining £600,000 will be offered to local community groups to improve organisations' skills through training and help them become more sustainable.
The government has also opened applications to the LGBT Advisory Panel, which will be made up of nine newly appointed members and three members already selected from charity Stonewall, the LGBT Foundation and the LGBT Consortium.
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Mordaunt said the money aimed to help people "live safe, happy and healthy lives where they can be themselves without fear of discrimination".
"Everyone in this country should feel safe and happy to be who they are, to love who they love, and to live their lives without judgment or fear," she said.
"That's why this government is stepping up its work to tackle bullying in schools, to protect more children and to stop hatred from festering and growing into discrimination in adulthood."
It follows the findings of the government's 2017 national survey of LGBT people, which led to its launch of the LGBT Action Plan in July 2018.
In September research published by Nottingham Trent University, Goldsmiths and the University of London found schools were not always sure how they should react to lesbian, gay, bisexual, transgender, queer and intersex (LGBTQI) parents, and needed to do more to make sure they felt welcomed.