Some genes on the X chromosome escape X-inactivation. Two copies of these genes are needed for normal development. These genes are also present on the Y chromosome. Hence, healthy males and females both have two copies of these genes. In Turner's, the SHOX gene seems to be one of the culprits, which is needed for normal skeletal development.
It appears that both of the X chromosomes are needed for normal development in females, as some genes on the X chromosome escape X-inactivation. Many of these genes are located at the ends of each arm of the X chromosome in areas known as the pseudoautosomal regions. Although many genes are unique to the X chromosome, genes in the pseudoautosomal regions are present on both sex chromosomes. As a result, men and women each have two functional copies of these genes. Many of these genes are essential for normal development (source: Genetics Home Reference).
Which genes on the X chromosome are responsible is not yet known, but a gene called SHOX seems to be involved, which is important for bone development and growth. The loss of one copy of this gene likely causes short stature and skeletal abnormalities in women with Turner syndrome (source: NIH), see Fig. 1.
SHOX is a transcription factor of the homeobox family. It acts during early embryonic development to control the formation of the development of the skeleton. It plays an important role in the growth and maturation of bones in the arms and legs source: NIH).
Fig. 1. Some developmental malformations seen in Turner's syndrome. source: University of Chicago
FYI - I wrote an answer before checking for duplicates. I think my answer does add additional information (the SHOX gene is not mentioned in the linked question's answer