When Sirius first got together with someone in third year, Remus was unexplainably protective. He constantly warned Sirius of everything and anything that could go wrong. At the time he put it down to the full moon but he knew full well it had nothing to do with that at all.
When Remus found Sirius with a girl from Hufflepuff in an empty corridor in their fourth year, he was angry. But he put it down to the fact that Sirius was breaking the rules, but he knew that that was not the problem.
In fifth year Remus caught Sirius leaving Hogsmeade with a boy from Ravenclaw and later that evening Sirius returned smirking.
Remus tried to convince himself that he wasn't jealous. He tried to convince himself that he didn't care, that he didn't mind, that Sirius could be with whoever he wanted. But the jealousy still bubbled up inside whenever Sirius talked about someone new.
Remus knew that he should be happy for Sirius, but he wasn't.

It was new, and it was scary. It was worse, in many ways, than turning into a monster once a month. At least he understood that. It made sense. Feelings didn't make sense. Especially when felt towards one of your best friends.
He wanted to hate Sirius, but he couldn't. He hated how comfortable Sirius was with himself. He hated that Sirius could easily talk about his new boyfriend or his new girlfriend. Remus wished he had the same confidence. But more than anything, he wanted to pretend that he didn't care, and for the most part, he could. It was just at night when everything became too overwhelming and he couldn't understand what he was feeling that he couldn't pretend anymore. But no one else knew that he was slowly losing control, and, to him, that was all that mattered.
Why didn't he care about James constantly pursuing Lily? Why didn't he mind if Peter had a relationship? What was it about Sirius' relationships that bothered him?

He didn't want to upset Sirius, so instead he just kept quiet, excusing himself whenever Sirius brought up the topic of his most recent relationship. The amount of times that Remus remembered some homework that he "hadn't finished yet" whenever Sirius brought up the topic should have been enough to draw attention, but it never was. James just assumed that Remus was genuinely going to revise. Sirius often didn't notice when Remus went quiet. Peter just thought that Remus was naturally quiet. And that was true, at least, in part.

The real truth was that Remus was jealous.