Cooperation in social dilemmas, such as public goods provision, is often undermined by the conflict between individual incentives and collective welfare. Social norms help resolve this conflict by aligning individual behavior with group expectations. In this study, we develop an evolutionary game-theoretic model that formalizes social norms through conditional expectations, meaning what individuals believe others will do based on prior cooperation levels. Agents earn game payoffs for participation in the game and social payoffs when their behavior matches the majority, whether cooperating or free riding. This setup models the interaction between empirical expectations and normative expectations, and shows that cooperation emerges when social rewards are sufficiently large. When social rewards are too small, cooperation fails to establish. We examine these theoretical insights through agent-based simulations, human experiments, and interactions with large language models (LLMs). Instructing LLMs to act as selfish individuals fails to sustain cooperation. However, when explicitly instructed to behave like fair-minded individuals, balancing self-interest with collective welfare, LLMs do succeed in establishing cooperation—particularly when social payoffs are large enough. This pattern reflects the behavior of human participants, who also tend to act as fair-minded agents under normative influence. Contrary to standard economic theory, which predicts self-interested behavior, both LLMs and humans behave as though cooperation is worthwhile when they expect others to cooperate and when social rewards are linked to conformity with group behavior. Our goal is not to demonstrate that LLMs can cooperate, but to use them as tools for testing theories of human cooperation. The results suggest that human cooperation aligns more closely with fair-minded conditional strategies than with purely self-interested rationality. These findings show how social norms, as conditional conformity, sustain cooperation across humans, artificial agents, and language models, challenging traditional economic assumptions.