In straight relationship story the conflict will usually end in a sacrifice of one of one of the couple while ensuring the other's safety.
In lesbian relationship the couple will usually get together or try to get together even if the consequences to the outside world are dire.
I think it's largely because innately, a straight relationship is part of a storytelling cycle - it's the foundation of a legacy, principally, the start of a family, a tangible symbol of the union of two characters. It's a component of the coming of age in the romantic sense: boy because man, man gets the woman, man and woman have children, responsibilities begin, and the cycle begins anew. At least in traditional storytelling, the gender difference means the male is the seeker of love, and has to earn it and the responsibilities and duty it entails, to prove himself worthy, where the female is the end goal, either in union or in sacrifice.
Gay relationships are more complicated to depict satisfyingly in that, dramatically, the story basically ends when the two characters end up together, because the family and all that entails is impossible. I've noticed some films, like Knock at the Cabin, play with idea of adoption, but storytellingwise it doesn't quite work because, ethics aside, an adopted or surrogate-born child is not the continuation of the union of two characters. Delaying or making the union of two gay characters as difficult or tumultuous as possible thus is usually the only real way to introduce conflict to the relationship. From what I see, in gay romance the conflict is largely about self-acceptance from one or both parties, which can result in success or failure, where with lesbian romance, especially nowdays, is often overtly feminist (hence depicted much more often), the emphasis being on defying male or patriarchal impositions and expectations, and that the purity or 'truth' of the love self-justifies whatever complications it causes.
And sadly, the reality is there is truth in stereotypes - even removing children from the equation, idealistic gay relationships are often hard for general audiences to accept because they're kind of rare. 'Experts' say it's about society and internalized homophobia and whatever but I think it's simply a product of what happens when two people of the same gender don't have to compromise on their monkey brain behavior - lesbians can end up indulging in the worst aspects of female socialization when dealing with each other, while gays often end up connecting over sex (or drugs) and little else.
