These are the only things I can come up with that feature both teen Sam and mentally ill Sam:

Nobody's Home (http://mentholpixie.livejournal.com/33708.html) by Menthol Pixie. Summary: All he knows is that he has to get away. (Cursed/amnesiac!Sam)
[This one is a cursed!Sam fic, but the curse does cause Sam to behave in ways that might be mistaken for mental illness]

You Better Start Swimming (https://www.fanfiction.net/s/8130931/1/You-Better-Start-Swimming) by Alex L. Kerr. Summary: Sam is abducted - kidnapped - for a month by a teacher who knows about the demon blood inside Sam. Sam and Dean struggle to stay afloat -and stick together- after the trauma. Anachronistic chapters, but still very easy to follow. Sam is 12, Dean 16/17. Warning: dark themes, see writer's notes.
[This one deals with Sam's mental state after being kidnapped. PTSD, possible Stockholm syndrome, suicide attempt, self harm]

Beneath the Trees 'verse (https://archiveofourown.org/series/18547) by Lise. Summary: Hunting wears you down. John gives the ultimatum. Sam makes a different choice.
[In which Sam doesn't go to Stanford. Severe depression and suicidal thoughts]

King of Infinite Space (http://archiveofourown.org/works/690081) by NezumiPi [Sadly, incomplete]. Summary: Sam and Dean are the sons of madman serial killer John Winchester. Dean pulled himself together, finished school and got a job. As for Sam, well, they say schizophrenia runs in families.

And if you don't mind self recs:

Collision Theory (http://archiveofourown.org/works/8507815/chapters/19497919). Summary: A few weeks before the first anniversary of their dad's death, Dean is called into the high school: Sam has finally crossed the line with his bad behaviour. Given a second chance, Sam must fulfil his punishment: the worst of which being weekly sessions with the school councillor, Mr Novak. Having recently started a new job in a new town far away from his old life, Cas tries to start afresh: two brothers collide with him, an impact he could never have seen coming.
[This is DeanCas, but it's not a major part of the plot. It mostly centres around Sam. Depression, PTSD, self harm, suicidal thoughts. WIP]

I hope there's something here you haven't read. Enjoy!

I agree with all of the ones lennelle listed, but I also have one more recommendation for you!

Open Book (https://www.fanfiction.net/s/11978995/1/Open-Book) by pineapplefan has a teenage Sam who struggles with mental illness. It's still a work in progress, but the author updates fairly regularly.

Hope it's what you're looking for!

Try sammehsayum's Baba O'Riley and Eleanor Rigby walk into a bar (http://sammehsayum.livejournal.com/4122.html)

Teenage Sam, depressed, coming out to his family, attempted suicide