Ultrasound elastography is a promising imaging technique that can assist in diagnosis of thyroid cancer. However, the complexity of the tissue movements under freehand compression requires the use of a parametric displacement model and a specific estimation method adapted to sub-pixel motion. Therefore, the aim of this study was to develop a motion estimation method for ultrasound elastography and test its performances compared to a classical block matching technique. The proposed method, referred to as Bilinear Deformable Block Matching (BDBM), uses a bilinear model with eight parameters for controlling the local mesh deformation. In addition, a technique of motion initialization based on a triangle scan of the images adapted to ultrasound elastography is proposed. The BDBM method includes an iterative multi-scale process. This iterative approach is shown to decrease the absolute error of the displacement estimation by a factor of 1.4 when passing from 1 to 2 iterations. The method was tested on simulated images and the results show that absolute displacement estimation error was reduced by a factor of 4 compared to classical block matching. We applied the BDBM method on three experimental sets of data. In the first data set, a phantom designed for ultrasound elastography was used. The two other sets of data involve the thyroid gland and were acquired using freehand tissue compression by ultrasound probe of a clinical ultrasound scanner modified for research. A similarity measurement based on local cross-correlation shows that, for experimental data, the BDBM method outperforms the usual block matching.