Why is Turner syndrome rarer than Klinefelter syndrome?
First of all, they are not caused by the same mechanism. They are both aneuploidies, but the mechanism is different.

Turner syndrome happens when one of the gametes (most commonly father's) lacks a copy of X chromosome, or the X chromosome is heavily damaged. This means that while normal women have two X chromosomes, Turner women have only one, and nothing else.

Klinefelter syndrome happens when one of the gametes retains an extra copy of the X chromosome. This way a Klinefelter man has two X chromosomes and Y chromosome instead of one X and one Y.

As you can see - the Turner woman is lacking some genetic information, while the Klinefelter man has too much of it. Generally, having too much genetic information is much easier to survive than not having enough of it, so this explains the relative difference on the most basic level.

But, you might say, what about Barr's bodies? Don't the women use only one X chromosome in each cell? And wasn't Y chromosome almost empty? 

Yes, these are an important consideration for these two syndromes. In each cell of a women's body one of the X chromosomes is inactivated and folded into a Barr's body. This inactivation is random, so part of the cells keep mother's and part of the cells keep father's X chromosome active. The trick is that the Barr body is not entirely inactive (Genes That Escape From X Inactivation). Only about 15% of the genes are active, but just the lack of them is mostly lethal (Turner fetuses are usually aborted, Wikipedia quotes olny 1% surviving until birth), and accounts for the health problems and deformations in Turner women.

A Klinefelter man has two X chromosomes in addition to his Y chromosome, so one of the chromosomes inactivates into a Barr body. The presence of Y chromosome directs his body to develop into a male pattern, but at the same time, the existence of Barr's body with those still active genes, causes his body to develop also feminine features. There is little to cause spontaneous abortion.