No, this is not true. In fact, this is a good example of how the media will pick up on a scientific study but then simplyfy the findings and mislead people.
Blone hair is “recessive” which means you need two copies of the blonde version of the gene to have that hair colour. You only need one copy of the brunette version, which is why people have darker hair than lighter hair. However, the blonde version of the gene will not die out unless there is some big disadvantage to being blonde (which there isn’t).
I think the popularity of hair bleach means we’re going to see a lot of blondes!
Hair contains two types of pigment, one that gives a dark colour and one that gives a ginger colour. Hair bleach destroys the pigment, but the dark one often goes first. That’s why when some people with really dark hair bleach it, they end up ginger. It’s not a giant conspiracy for ginger haired people to take over the world, it’s just chemistry!