Research carried out by the Fostering Network suggests almost half of fostered young people are already living with their third foster family since going into care.
The group has warned that 750 more foster carers are "urgently" needed to meet the demands of the care system.
It urged people to "open their hearts and homes" to vulnerable youngsters.
Currently, more than 5,500 children are in foster care in Scotland, living with 4,400 families and carers.
The Fostering Network surveyed 250 children, teenagers and foster carers across Scotland and discovered that many young people had failed to find stability.
Almost half were already living with their third family, a quarter were with their fourth family and about 20 were living with their 10th family since going into care.
There was a particular need for homes to be found for vulnerable teenagers, siblings and disabled children, the study found.
Carla, 23, was taken into care at the age of 12 and had eight foster homes before moving in with the Randalls.
"Looking back now I realised that the Randalls saved my life," she said. "I never understood the extent of the neglect and abuse I had endured until I came to live with a 'normal' loving family.
"They were just always themselves, the smallest details meant so much to me.
"They nurtured a young, angry, untrusting teenager to become a positive, empathetic and successful young woman."
The Fostering Network said instability had a detrimental effect on the child's education and wellbeing, while finding a stable foster carer from the outset could lead to improved relationships and a happier childhood.
Sara Lurie, director of the Fostering Network Scotland, said: "As each year passes, we see more and more children coming into care.
"We need people who can open their heart, and their homes, to vulnerable children and young people and use their skills to help support them to reach their full potential.
"In particular we need people who have the skills, patience and passion to look after teenagers who may have had a really tough time and be facing some real challenges, and to offer them love, stability and security.
"A good foster carer will believe in the ambition of the children in their care in the same way they'd believe in the ambition of their biological family members."
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A Scottish government spokeswoman said: "Giving young people security is paramount and we have done a great deal of work with our partners across local government and the third sector to improve how we intervene early when there is a problem within families to find appropriate solutions quickly.
"We have also expanded the age at which young people can remain in foster care as part of the continuing care provisions and the support available when they transition into independent living."

Andrew Steele, from Dunfermline, turned at a police station and confessed to and raping one girl and touching another between 1996 and 2005.
He was initially released pending further investigations but was detained months later. He then said he been hallucinating when he made the confession.
Sentence was deferred for background reports.
The High Court in Edinburgh heard that Steele, aged 49, told officers his mother was "up in the sky" urging him to confess to the sex crimes when he made his revelations.
Police traced the girls and took statements from them. Both victims were aged seven when the abuse began.
Steele admitted rape, indecency offences and failing to turn up for an earlier court date, when he appeared at the High Court in Edinburgh.
Advocate depute Mark McGuire said: "On 28 December in 2014 the accused voluntarily attended at Kirkcaldy police station and disclosed that he had interfered with the girls."
"On 3 March in 2015 the accused was detained. Under caution he told the officers that he was having hallucinations on the day that he confessed. He told them that his mother was up in the sky telling him to confess. He maintained this position throughout the interview."
The prosecutor said before his first appearance at the police station Steele had gone to hospital and claimed he was hallucinating.
"He was assessed by three medical professionals who concluded that he was suffering from alcohol withdrawal. He was deemed fit to be discharged," said the prosecutor
Steele was remanded in custody and put on the sex offenders register.
Teenagers in foster care in Scotland are being moved too often, according to a campaign group.
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A man has admitted sexually abusing two girls in Fife over a nine year period.