A mother is said to have given her 13-year-old daughter three ecstasy tablets as a birthday present, the BBC learned.
Middlesbrough mayor Ray Mallon highlighted the case when speaking to Look North about hundreds of troubled families being supported on Teesside.
The investigation also found Middlesbrough Council was spending £600,000 per year on a family with six children in care.
Children should be targeted "in the womb" to stop problems, Mr Mallon said.
He said that children were being let down by parents who abused alcohol and drugs.
"There was one where a mother gave a 13-year-old child some ecstasy tablets, hard drugs, for her birthday," the mayor said. "I just despair."
The BBC was told that a troubled family with six children under the age of 13 was costing Middlesbrough Council more than £600,000 per year.
Three of the children were in foster care at a cost of £135,000, while the remainder were in residential care at a cost of £468,000.
Mr Mallon said that as an ex-police officer, he recognised the same family names getting involved in crime year after year.
"We should target children whilst they're in the womb," he said, "because it's clear that you can work out by the parents which kids are likely to have problems."
Prime Minister David Cameron appointed Louise Casey to lead a Troubled Families Unit in 2010.
She said Middlesbrough had "a way to go" but was making some progress.
Ms Casey said it was right that councils supported parents through a programme of "family intervention".
"The most important thing about this job - whether it's mayor Mallon, whether it's the family intervention worker or me - is determination," she said. "You have to be determined to get this right."