We consider online bulk-delete operations on a large database table organized as a primary (sparse) B+-tree index on a multi-attribute key. Using the natural range partitions induced by prefixes of the key, we define a multi-granular key-range locking protocol in which a bulk operation locks a small number of logical fragments of the table covering the target of the operation. We also present an efficient and recoverable bulk-delete algorithm that minimizes the work needed in B-tree rebalancing and in transaction rollback. All the locks needed for a bulk-delete operation are acquired during a scan of the leaf pages covering the target key range; in this scan the records qualifying for deletion are only marked as deleted. The records are physically deleted in a rebalance phase that avoids visiting subtrees in which all records qualify for deletion, thus saving considerably on the number of rebalancing operations.