The human-centered word association test (WAT) serves as a cognitive proxy, revealing sociocultural variations through lexical-semantic patterns. We extend this test into an LLM-adaptive, free-relation task to assess the alignment of large language models (LLMs) with cross-cultural cognition. To mitigate the culture preference, we propose CultureSteer, an innovative approach that integrates a culture-aware steering mechanism to guide semantic representations toward culturally specific spaces. Experiments show that current LLMs exhibit significant bias toward Western cultural (notably in American) schemas at the word association level. In contrast, our model substantially improves cross-cultural alignment, surpassing prompt-based methods in capturing diverse semantic associations. Further validation on culture-sensitive downstream tasks confirms its efficacy in fostering cognitive alignment across cultures. This work contributes a novel methodological paradigm for enhancing cultural awareness in LLMs, advancing the development of more inclusive language technologies.