Actually you are just a little off. There is a real Major Crimes Unit in the LAPD. I have attached photos for you to look at from the LAPD website.
In the first pic, you can see the Chief of Staff is at the top, and then the Professional Standards Bureau and so on until you reach the CTSOB, or the Counter-Terrorism and Special Operations Bureau. In that section is where the Major Crimes Division sits, along with Emergency Services Division. This is why Major Crimes is called in not only for high profile murders, but terrorism threats, serial killers, and kidnappings.
Now in the second picture, you see that the Commanding Officer of the CTSOB is Deputy Chief Michael Downing. He reports directly to Assistant Chief Michel R. Moore from the Office of Special Operations. Now when I perused the command Staff of the LAPD, there are SEVERAL Captains, some Commanders, and about a dozen Chiefs (Deputy or Assistant). Here is the link to the command structure - http://www.lapdonline.org/lapd_command_staff
You will notice in that link that SEVERAL of the Captains are women who are quite young. Some look in their 30s even. So that whole “went up the ladder fast to make Captain” doesn’t seem so rare. However, it does look rare for a woman to be a Commander or higher.
Now, in the third picture, you will see the Major Crimes Division. Captain Steven Sambar is the Commanding Officer of this division. So, if you were to fit this to the show, Captain Sambar is Captain Raydor, Assistant Chief Moore is Assistant Chief Taylor, and Deputy Chief Downing is Deputy Chief Pope (remember that he was going to that position at the end of The Closer? That is why Taylor was made Assistant Chief in the first place - to take his place. When Pope was Assistant Chief, we saw him all the time, but the Deputy Chief is hardly seen as he is in charge of the Office of all Special Operations. It is a huge section.
So, you see, Sharon could be made Commander. However, there looks to be only one in each division, and because Major Crimes is only one part of a bigger division, that spot might be filled right now. As I said, there are a whole lot of Captains, but after that, it is about a dozen Commanders and Chiefs, and there are about a dozen different sections in the LAPD. Sometimes, the only way to go up is if someone dies or retires. I hate to put it that way because it sounds so morbid, but it is actually the truth, because you can’t have more than one Commander or Chief in one section.
(Oh and all the information I got was from the LAPD website and my brother, who actually is a cop :p). Hope this helps ♥