Linear sketching algorithms have been widely used for processing large-scale distributed and streaming datasets. Their popularity is largely due to the fact that linear sketches can be naturally composed in the distributed model and be efficiently updated in the streaming model. The errors of linear sketches are typically expressed in terms of the sum of coordinates of the input vector excluding those largest ones, or, the mass on the tail of the vector. Thus, the precondition for these algorithms to perform well is that the mass on the tail is small, which is, however, not always the case -- in many real-world datasets the coordinates of the input vector have a {\em bias}, which will generate a large mass on the tail. In this paper we propose linear sketches that are {\em bias-aware}. We rigorously prove that they achieve strictly better error guarantees than the corresponding existing sketches, and demonstrate their practicality and superiority via an extensive experimental evaluation on both real and synthetic datasets.