Large Language Models (LLMs) have achieved remarkable success in Natural Language Processing (NLP), yet their cross-lingual consistency remains a significant challenge. This paper introduces a novel methodology for efficiently identifying inherent cross-lingual weaknesses in LLMs. Our approach leverages beam search and LLM-based simulation to generate bilingual question pairs that expose performance discrepancies between English and target languages. We construct a new dataset of over 6,000 bilingual pairs across 16 languages using this methodology, demonstrating its effectiveness in revealing weaknesses even in state-of-the-art models. The extensive experiments demonstrate that our method precisely and cost-effectively pinpoints cross-lingual weaknesses, consistently revealing over 50% accuracy drops in target languages across a wide range of models. Moreover, further experiments investigate the relationship between linguistic similarity and cross-lingual weaknesses, revealing that linguistically related languages share similar performance patterns and benefit from targeted post-training. Code is available at https://github.com/xzx34/Cross-Lingual-Pitfalls.