Freedom of Information requests by the party show hundreds of the youngsters seen by health boards’ weight management services since 2014/15 were aged between two and four.
The Tories said the figures highlighted the crucial need to tackle obesity.
The statistics show that between 2014/15 and 2016/17, health boards saw at least 5,129 weight management referrals for under 18s.
Some boards were unable to provide data, therefore the figures are likely to be higher.
There were at least 1,600 referrals last year, while over the three-year period, more than 500 children seen by weight management services were aged between two and four.
In addition, more than 60,000 adults were referred over the three-year period.
Tory health spokesman Miles Briggs said: “Obesity is going to be one of the great public health challenges of our time.
“These figures show tens of thousands of people - even children as young as two - are being referred to specialists.
“If we don’t take steps to tackle this problem now, the consequences for both the health of our population and the future strain on the NHS will be severe.”
The Scottish Government has been consulting on proposals for improving diet and weight in Scotland.
Mr Briggs added: “The Scottish Government’s consultation on obesity has now closed, and it’s essential that - at the other end of this process - we have a meaningful strategy to improve diets and healthy lifestyles.
“It’s not all about what the NHS and government do - there has to be an element of personal responsibility too - but from a health policy perspective, we cannot afford to get this wrong.
“It is also vital that we see cross-portfolio working on this from SNP ministers as many of the changes and interventions will need to be educational as well as health related.”
Public Health Minister Aileen Campbell said: “We put forward a bold package of measures in our draft diet and healthy weight strategy to help everyone make positive choices, empower personal change, and makes Scotland a healthier place to live.
“The strategy includes world leading proposals to restrict the marketing of foods high in fat, salt and sugar.
“Now that the strategy consultation has closed, we will continue to lead the consensus across the political spectrum as we set out our next steps.
“As was the case with tobacco and minimum unit pricing for alcohol, when Parliament unites around a common purpose we can make a real and lasting change to the health of our nation.”