As the title,
I'm a player from China, mainland.
The lore and story which we can see in mainland is not so much as in here or in Taiwan,
as Tencent wouldn't give out so much stories.
But I loved the stories of Journal of Justice, and the Judgement. I found the Chinese version in forums of Taiwan, and shared them to my friends.

But when I told them stories in the JOJ, they just tell me JOJ and Judgement are only fan fictions.
This made me so unhappy. I don't think JOJ are fan fictions, and I need evidence to prove myself !!

P.S. how can I get to the JOJ pages through the official website of LOL? I searched for the whole website and I am not able to find it.

The Journal of Justice and Official League Judgements were written by Riot, and are canon, not fan-fiction so rest easy.
There are fan-fiction judgements made by players on the forums, several of them are a good read, but shouldn't be confused with the official ones. A section containing the official League Judgements can be found in Sylph's Chamber of Antiquity. An easy way to tell whether a Judgement is official or fan-made is looking at the user name, Riot staff usernames are usually coloured red and have "Riot" before the actual username, hence the nickname "Reds".

The official JoJ site is http://na.leagueoflegends.com/story/...13-july-20-cle, it's a section of the official site, written by Riot staff.

All of the Lore based data can be found on the North America server, other regions are like you said somewhat limited to what is given out (probably due to translation issues and the like) but there is a great deal there to look at.

I'm sure that this should be proof enough for your friends, good luck convincing them.