Entity linking (EL) for the rapidly growing short text (e.g. search queries and news titles) is critical to industrial applications. Most existing approaches relying on adequate context for long text EL are not effective for the concise and sparse short text. In this paper, we propose a novel framework called Multi-turn Multiple-choice Machine reading comprehension (M3) to solve the short text EL from a new perspective: a query is generated for each ambiguous mention exploiting its surrounding context, and an option selection module is employed to identify the golden entity from candidates using the query. In this way, M3 framework sufficiently interacts limited context with candidate entities during the encoding process, as well as implicitly considers the dissimilarities inside the candidate bunch in the selection stage. In addition, we design a two-stage verifier incorporated into M3 to address the commonly existed unlinkable problem in short text. To further consider the topical coherence and interdependence among referred entities, M3 leverages a multi-turn fashion to deal with mentions in a sequence manner by retrospecting historical cues. Evaluation shows that our M3 framework achieves the state-of-the-art performance on five Chinese and English datasets for the real-world short text EL.