Godzilla did not lose to Mothra in their first fight in the 1964 film, Mothra vs. Godzilla. A common misconception about Toho's movies is this idea that in their first-ever encounter, Mothra actually defeated Godzilla, but that's not exactly what happened, considering Mothra died in that battle.
Mothra and Godzilla have decades of history. The two have appeared in several movies together, and the status of the relationship usually depends on Godzilla's role, especially since Godzilla: King of the Monsters.
Mothra, three years after her solo movie, battled Godzilla for the first time in 1964's Mothra vs. Godzilla (not to be confused with 1992's Godzilla vs. Mothra). When Godzilla was terrorizing Japan, it was up to Mothra to oppose Godzilla. In their battle, Mothra gave the King of the Monsters a bit of trouble by dragging him by his tail and keeping him at bay with her silk, but Godzilla - as usual - was relentless in his assault. Mothra held out for a while, but couldn't stand up to Godzilla's atomic breath, which killed Mothra. After the fight, Mothra's eggs hatched, and two Mothra larvae went up against Godzilla. With their silk, the two were able to bind him up in a tight ball of silk, rendering him immobile.
Over the years, Godzilla's ultimate defeat has led to the notion that he lost to Mothra, but it's important to recognize that the fully-grown Mothra actually died to Godzilla. Despite her efforts to save Japan, Godzilla ultimately triumphed over the creature. It took, not one, but two Mothra larvae to tie up Godzilla, who was far too powerful to be stopped physically. In the end, Godzilla was indeed stopped, but this can hardly be called a fair fight.
This wasn't the last time Mothra was shown to be incapable of beating Godzilla without help. Another notable example occurred in the 1992 rematch, Mothra even had to be upgraded with a stinger in Godzilla: King of the Monsters in order to properly defend herself against predators like Rodan.
