Why doesn't ovulation occur in both ovaries simultaneously?
Each ovary has hundreds of thousands of follicles. And, of course, each one of those follicles is exposed to the same blood supply.
Therefore, your question...

Why isn't ovulation seen in both ovaries together?

... is the same of:

Why isn't ovulation seen in two follicles (regardless the ovary)?

All follicles are stimulated by FSH and they indeed start to mature at the same time. However, in a given cycle, normally only one follicle fully grows1. What we have here is a competition among follicles.
Guyton is not my favourite book on physiology, but it suggests an explanation for this:

Only One Follicle Fully Matures Each Month, and the Remainder Undergo Atresia:
After a week or more of growth — but before ovulation occurs — one of the follicles begins to outgrow all the others; the remaining [...] developing follicles involute (a process called atresia), and these follicles are said to become atretic. The cause of the atresia is unknown, but it has been postulated to be the following: The large amounts of estrogen from the most rapidly growing follicle act on the hypothalamus to depress further enhancement of FSH secretion by the anterior pituitary gland, in this way blocking further growth of the less well developed follicles. Therefore, the largest follicle continues to grow because of its intrinsic positive feedback effects, while all the other follicles stop growing and actually involute.
This process of atresia is important, because it normally allows only one of the follicles to grow large enough each month to ovulate; this usually prevents more than one child from developing with each pregnancy. The single follicle reaches a diameter of 1 to 1.5 centimeters at the time of ovulation and is called the mature follicle.

Therefore, the reason why we don't see both ovaries ovulating at the same time is the same reason why we don't see one ovary ovulating twice at the same time: the fastest growing follicle inhibits the growth of the other follicles (in the same ovary or in the contralateral ovary, it doesn't matter).
Besides that, it also explains the pattern of ovulation: in a given cycle, the left and the right ovary have equal chances to ovulate. Thus, the ovaries don't take turns like this (L is left and R is right):
L → R → L → R → L → R → L → R → L → R → L → R → L → R → L → R → L → R etc...
The real pattern is probably random:
L → L → R → R → L → R → R → R → L → R → L → R → L → L → R → L → L → L ETC...
According to Ecochard (2000):

It is concluded that in normally fertile women, the cycle length and the hormonal profile are independent of the, most probably random, site of ovulation. (emphasis mine)

Sources:

Hall, J. and Guyton, A. (2011). Textbook of medical physiology. Philadelphia, Pa.: Saunders.
Ecochard, R. (2000). Side of ovulation and cycle characteristics in normally fertile women. Human Reproduction, 15(4), pp.752-755.


1Note that, for answering this question, we will (wrongly) assume that there is no polyovulation in humans. However, as polyovulation does occur in humans, ovulation can sometimes be seen in both ovaries.