How common is intinction in the Catholic Church? Intinction at communion is "the practice of dipping the bread in the cup and partaking the elements simultaneously". 

The Presbyterian Church in America (PCA) at their 2012 General Assembly narrowly voted in favor of an amended version of Overture 30, which would prohibit celebrating communion by intinction. (But this will not actually happen unless 2/3 of the presbyteries, and the next GA, also approve it.) Many lengthy arguments were deployed on both sides.

One of the minor arguments on the pro-intinction side is that the practice is common among Catholics and the Orthodox, and it would be a welcoming gesture if this mode of celebration was explicitly authorized. There are also other Protestant denominations, including Presbyterians, that allow it among other options. I believe that this is indeed the standard way of administering communion in the Eastern Orthodox churches, but I'm not so sure about the Catholics (specifically, Catholics in the USA). The General Instruction of the Roman Missal allows it, but the language seems like it's being authorized only as an alternative method.

How common is intinction in Catholic churches in the USA? Is it seen as a relatively normal thing to do, or is it controversial?
All Catholic churches I have attended have you line up to the front, you are given a communion (either placed in your hand or in your mouth), and on the way back to your seat is a wine bearer, who will offer you a sip from the cup. Unless you place the host in your mouth and then partake from the wine cup, it would not be possible to do this.

In other churches, as you have said, this can be common, but at a standard Catholic church it's not frowned upon as much as not possible following the standard setup.