Four-circle X-ray diffraction measurements down to 15K on single crystals of the title superconducting salts (BEDT-TTF)2X have revealed an anomalous temperature dependence of the lattice parameters in a broad range of temperatures centered around 100K, where many anomalies have been observed such as in transport, magnetic and optical properties in both materials. These prominent lattice anomalies, which are clearly featured in broad peaks of the interlayer spacing for X=Cu(NCS)2 and the a-axis (parallel to polymeric chain) for CuN(CN)2Br, are likely to be associated with the respective conformational ordering of the terminal ethylenes. The other lattice parameters change with temperature in a way that the unit cell volume of each compound varies only smoothly with temperature. From the estimated volume expansion coefficient in combination of available specific heat and compressibility data, we evaluate the low-temperature Grueneisen parameter to be 5-6 for both compounds. A thermodynamical analysis within a Bardeen-Cooper-Schrieffer theoretical framework concludes that the well-known high sensitivity of Tc to hydrostatic pressure is caused by the highly volume-sensitive electron-phonon coupling constant, and that the strong lattice anharmonicity plays an essential role in the superconductivity.