Sofia Legg's family was told there was a six-month waiting list for therapy despite an urgent GP referral for help.
The inquest coroner in Taunton recorded a conclusion the 14-year-old had taken her own life by hanging.
Somerset Partnership NHS Foundation Trust said it had apologised to the family and improvements had been made.
The inquest heard Sofia, a student at Kings of Wessex Academy in Cheddar, had been suffering from low moods, was self-harming and was expressing thoughts about taking her own life.
She was referred to the Child and Adolescent Mental Services (CAMHS) in March 2015, but did not meet the criteria for specialist support.
In February 2016, her GP made an urgent referral to CAMHS, after Sofia confessed to having suicidal thoughts.
But after being seen twice by CAMHS in July 2016, she was told there was a six-month waiting list for cognitive behaviour therapy.
Her family found her body when they returned home on the evening of 20 September.
On the day before she died, a CAMHS worker met Sofia and her mother, giving a crisis plan stating her mother Sandy Legg needed "to watch her and keep a close eye on her".
But investigation reports stated Sofia had to be "supervised at all times".
"If constant supervision was necessary then this should have been stated directly to me before I left with Sofia," her mother told the hearing.
Speaking after the inquest she added: "As a family we will forever mourn the loss of our kind, bright, beautiful daughter.
"Sofia was failed by the system and we, as a family, feel badly let down."
Somerset coroner Tony Williams said he would be following up his concerns about the death with the trust.
Mental health services failed a teenage girl who had suicidal thoughts, her family has said following an inquest into her death.