Its a misconception
Most people argument with the cost of fixing a known error at design time.
If you fix an error at design time, you don't know it. You have to think about all the aspects of the design, which is very time intensive.
And its a tricky task, since some aspects of the task may be unknown. Like: How many visitors? What's the real performance of the plattform under pressure?
I will not claim any number, but i don't think, that making and veryfying a very detailed design can be done so fast, that it saves time hundredfold.
This will work only in retrospective, when you know, where the design failed.
Its not easy to find the right balance between too detailed and too rough a plan.
But if you do a new task, some errors will go through all phases and pop up at the customer.
A 100% error free project may happen, if you are adapting a product, but even this seems unlikely to me, if you have to deliver at a reasonable cost-
The amount of testing an redesigning in building a mars rover, will not be paid in the industry for a production machine.
They want this only until you tell the estimated cost..