The one that always gets me isn't the big, dramatic blow-up; it's the quiet, exhausted moments after a major decision where the distance between them is palpable. Like in season 2, right after they irradiate Level Five in Mount Weather. They just saved everyone, but the scene in the hallway is so heavy. They're standing there, covered in blood and sweat, and Clarke can barely look at him. Bellamy's trying to connect, to find some shared ground in the victory, but she's already a million miles away, disappearing into the weight of being 'the one who bears it.' His conflict is wanting to pull her back, to share the burden, and hers is believing she has to carry it alone so he can stay 'good.' That emotional conflict isn't about anger; it's about two people who love each other choosing completely different methods of survival and creating this awful rift.
Another brutal one is in season 4 when they're arguing in the throne room about the list for the bunker. Bellamy wants to save everyone, the idealist turned realist who's now circling back to hope because of Clarke's earlier influence. Clarke is being the cold, practical leader he taught her to be in season 1. She's using his own old logic against him, and you can see it wound him. He shouts, 'You're being irrational!' and it's so raw because he's not really talking about the list. He's begging her not to lose the part of herself that fought for humanity, that fought for him. The conflict is this painful role reversal where they've each absorbed the other's worst traits for survival, and they're horrified to see it mirrored back.
The moments that highlight their conflicts best are often the ones without resolution—where they just have to walk away from each other because the chasm is too wide. It's that specific, aching tension that makes the rare moments of alignment so powerful. You live for those brief seconds where they're on the same page again.