Exclusive breastfeeding does suppress ovulation for many women in the first months, and this is a real biological effect rather than a myth in itself.
It stops being dependable quickly. The suppression relies on frequent feeding around the clock, including at night. Once a baby sleeps longer, takes any formula, or starts solids, the protection weakens — and ovulation returns before the first period, so there is no warning that fertility has come back.
That is the part that surprises people: you can conceive without having had a single period since giving birth.
If another pregnancy soon is not what you want, talk to a doctor or midwife about contraception that is compatible with breastfeeding. Several options are.