A fifth of babies are being put at risk of cot death, often due to concerns over "flat head syndrome", experts say.
Putting a baby to sleep on its back reduces cot death risk
The Foundation for the Study of Infant Deaths poll say the figure is double that seen six years ago.
A failure to put babies on their backs increases the danger of cot death by nine times.
Concerns over flat head syndrome, or plagiocephaly, are increasing, but FSID says it can be avoided if people let babies play on their fronts.
FSID fears publicity about flat head syndrome is increasing.
But it says the condition is "entirely cosmetic, almost always corrects itself within a year, and may be avoided if parents give the baby plenty of awake time on the front or sitting up".
The latest figures, from 2004, show there were 329 cot deaths in the UK, down just 10 from 2000.
FSID has now produced a new leaflet "Sleep on the back, play on the front", highlighting the safe-sleep message, and explaining the importance of supervised front play.
Its study of 1,545 mothers with babies under a year old found almost one in five mothers (19%) with babies aged under six months never give their babies front play time, the researchers found.
Just 22% regularly give their babies time to play on the front, although 63% of mothers with children aged 0-6 months regularly give them time to play on their back on the floor.
FSID reports paediatric physiotherapists highlighting an increase in the number of children who show a delay in movement development.
Joyce Epstein, FSID's director, said the new research was alarming: "Our fear is that the lifesaving message to sleep babies on the back to reduce the risk of cot death will be undermined by a mistaken perception that flattened heads poses a greater danger. It does not."
Clare Jolly, health visitor advisor to FSID, said flat head syndrome does not do any medical harm, and does not need to be rectified with expensive treatments or devices.
"We are seeing more babies with flattened heads because they are spending so much of their waking time lying flat on the back.
"Parents often wrongly think they should not let their baby be on the front at all.
"When a baby is awake they should enjoy different positions from the very beginning."
Peta Smith, vice chairwoman of the Association of Paediatric Chartered Physiotherapists , said: "The first few months of life are an important time for babies to start to become aware of their bodies and develop the skills they require for rolling over, sitting and crawling.
"Simple measures like giving your baby supervised tummy time every day will help them co-ordinate, balance and control their body and give them a foundation for all movement and skills."