I work with civil rights for children and youth and am trying to create a website for LGBT youth and children and their families. The website needs to be safe as it is still very dangerous in many countries to admit to being gay (or LGBTQ).

I have a series of short children's films that I would like to make accessible for free on a website but the youth, families and teachers who access the site need to be untraceable. 

Can I make a website where the security lies within the website?
People and kids coming to the site will not be setting up their own security measures to not be traced.

Don't do it

If you are targeting unsafe regimes, the consequences can be severe. For example, in Iran, condemned homosexuals are usually sentenced to death by hanging.

It would only take the slightest lapse on the part of your site, or one of your users for this to be incriminating evidence. Can you imagine the guilt you would feel that your well-intentioned site ended up being evidence that led to an execution?

To operate online subversively in a repressive regime takes extreme skill and caution. Simply using Tor would be a red flag in itself. While hacker groups like anonymous may be able to get away with this, normal users cannot.

There have been comments on other answers saying (roughly) "use HTTPS, it's fine". I cannot stress strongly enough how that is dangerous advice. HTTPS reveals the domain name the user is accessing, it leaves trace on the client computer, and it's likely that nation-states (including Iran) can produce fake certificates and intercept all HTTPS traffic.
