What type of beer would the pilgrims have had access to in the early 1600's, and if they did have beer, would it be a good pairing with turkey?
Also bear in mind that beers back then were made out of lots of ingredients that we'd think of as odd. Cock ale was obviously a thing, which was beer that was fermented (or boiled) with a rooster in it. Lots of beers would also have been strained through spruce post-boil which would've given them some wood or pine quality. Lots of other flowers were often involved.
Check out George Washington's small beer recipe here. Though this is a poor transcription and a lot of folks agree that it should read "Bran, Hops to taste", it's still clear that the grain is sort of a "for taste, whatever you can spare" thing while the molasses is the star of the show. A lot of early drinks were like that, involving some sort of concentrated or burnt sugar in addition to grains, which generally weren't barley at the time.
However, in the most ideal case where you had access to the best beer you could find...it would be some form of brown ale or lighter porter. Maybe an amber ale if you could find one that wasn't very hoppy. The stable kilning process that enabled people to make pale malt wasn't really invented until the mid-1600's and wasn't very popular until the 1700s, so brewers in the 1600s would've used a sort of generic-y brown malt that might taste slightly burnt and premium beers may have been able to include some lighter malt that was much harder to make and find. And it'd probably still have some molasses in it.
Beer was commonly drunk in the Elizabethan period in Europe, but what they drank is different from the beer we're used to today.
Beer and ale, being grain-based, were important dietary staples -- it's said that beer is liquid bread, and that's not far off. For the common man (not nobility), in particular, grain made up a substantial part of the diet, with meat being fairly rare.
Common beer was not aged for months or years like some beers today; rather, a batch might be produced in as little as half a week. These are "small beers" (or "small ales", for the unhopped variety), which are mildly alcoholic but drinkable in volume without unfortunate effects. These small beers/ales were produced in the home/manor; it was just one more task for the cooks. See, for example, Markham's The English Housewife, 1615. (I don't know of an online copy, sorry.)
Note that this means that the pilgrims wouldn't have brought beer from Europe; it'd be consumed, or probably go bad, before they reached the new world. So they probably didn't have beer in their first year because they'd need to wait for a grain harvest. After that, if they brought hops with them then they could have made small beers like from the old country; if they didn't bring hops, they probably made ales instead.
Turkey would have been new to their palates (it's a new-world bird), but beer was commonly drunk with meals where they came from, and those meals sometimes included other fowl. So, probably it pairs fine. However, they probably didn't care as much about the proper pairing as we might today, the same as we might not care about pairing the proper cola with the turkey sandwich we have for lunch.
You may find the following helpful:
The bulk of what they drank would have been small beer: very low alcohol content beer. It was often unfiltered. This was more or less consumed daily instead of water, since there was no sanitation at the time.