Wealthy young socialite Betty Kane was completely infatuated with Robin the Boy Wonder that she ultimately created a costumed identity to become his unoffical partner in order to get close to him depite the extreme danger of her romantic endeavor.
Like her aunt who had adopted the costumed identity of Batwoman in order to get close to Batman, young Betty was likewise deeply attracted to the young Robin and created the costumed identity of Bat-Girl to attract the young crime-fighter. While Robin returned her affections, nothing ever really came of the relationship, most likely due to the controlling influence of Batman and the fact that they usually engaged in the very dangerous practice of open combat. It was probably the extreme danger and death of her aunt Katherine who as the orignal Batwoman was murdered because of her association with Batman that influenced young Betty to basically abandon her career as Bat-girl. Betty was succeeded in the role of Batgirl by the young daughter of Police Commissioner Gordon who Batman reluctantly took into his confidence.