Edge computing has low transmission delay and unites more agile interconnected devices spread across geographies, which enables cloud-edge storage more suitable for distributed data sharing. This paper proposes a trustworthy and reliable multi-keyword search (TRMS) in blockchain-assisted cloud-edge storage, where data users can choose a faster search based on edge servers or a wider search based on cloud servers. To acquire trustworthy search results and find reliable servers, the blockchain-based smart contract is introduced in our scheme, which will execute the search algorithm and update the score-based trust management model. In this way, search results and trust scores will be published and recorded on the blockchain. By checking search results, data users can judge whether the returned documents are top-k documents. Based on the trust management model, we can punish the malicious behavior of search servers, while data users can choose more reliable servers based on trust scores. To improve efficiency, we design a threshold-based depth-first search algorithm. Extensive experiments are simulated on Hyperledger Fabric v2.4.1, which demonstrate our scheme (with 16 threads) can reduce the time cost of index construction by 92% and the time cost of search by 82%, approximately. Security analysis and extensive experiments can prove the security and efficiency of the proposed scheme.