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."
Teenagers in foster care in Scotland are being moved too often, according to a campaign group.