Flexible intelligent metasurface (FIM) has recently received considerable interest due to its advantage in realizing a better channel condition by dynamically morphing its surface shape. An FIM consists of multiple elements deposited on a flexible substrate. These elements can not only transmit signals, but also adapt their displacements in a direction perpendicular to the FIM surface via an attached controller. In this paper, we consider the channel estimation problem for the uplink of an FIM-enhanced communication system via customizing the orthogonal matching pursuit (OMP) method. Specifically, we formulate an optimization problem of minimizing the column coherence of the measurement matrix by optimizing the FIM's surface shape, subject to the morphing range constraint. Based on the estimated direction of arrival (DOA) and channel gain, we further investigate the signal-to-noise ratio (SNR) improvement in the FIM-enhanced downlink multiple-input single-output (MISO) system. Numerical results demonstrate that an FIM significantly outperforms a conventional rigid uniform planar array (UPA), thereby showing that FIM can substantially improve channel estimation accuracy and achieve SNR improvement, even when using estimated channel parameters.