Are turkeys and chickens more closely related to each other than either are to, say, ostriches? penguins? grouse? Is there any other bird that is more closely related to either? Or, if they’re not all that closely related, just how far apart are they?
They are both members of family Phasianidae, so yes, they are more closely related than to ostriches and penguins, but grouse are in that family, as are pheasants, quail, peafowl and bobwhites:
http://i-bird.com/Family/Phasianidae.htm
The chicken is genus Gallus (junglefowl), and there are several wild junglefowl species, including the red junglefowl (Gallus gallus) which is the ancestor of the domestic chicken.
The turkey is genus Meleagris.
I hope the resident bird experts don’t grouse to much about me poaching on their territory …
Chickens and turkeys are pretty closely related, but are not each others closest relatives.
The order Galliformes (Chicken-like birds), besides the chickens, pheasants, quail, turkeys and grouse, also contains the Megapodes (also known as Brush Turkeys or Mound-nest Builders), family Megapodidae, of the Australian region; the Guans and Curassows, family Cracidae, of the New World tropics, and the Guineafowl, family Numididae; of Africa.
The chickens, pheasants, quail, turkeys and grouse are all rather closely related, and there have been differing opinions over whether to lump them all in the same family (Phasianidae), or to split the most distinctive types into separate families.
Traditionally, the grouse were often split off as family Tetraonidae and the turkeys as family Meleagridae. More recent genetic analysis suggests these should be included in the Phasianidae, but that the New World Quail (including Bobwhites) are different enough to get their own family, Odontophoridae.
Among the Phasianidae, the two species of turkey, the Wild Turkey (from which the domestic version descends) and the Ocellated Turkey (of Mexico and Guatemala) are evidently most closely related to the grouse. The several species of junglefowl (including the Red Junglefowl, from which the domestic chicken is descended, as yabob has noted) are more closely related to pheasants such as the familiar Ring-necked Pheasant.
The chicken-like birds are one of the more primitive groups among birds. Their closest relatives appear to be the duck-like birds, order Anatiformes, including ducks, geese, swans, etc. Neither of these groups is particularly closely related to ostriches, penguins, or other groups.
I just did a little more checking, and it looks like there is still a lot of dispute about classification within the Phasianidae. Another reference says turkeys are most closely related to pheasants, while chickens are most closely related to partridges and Old World quail. Suffice to say turkeys and chickens are related, but not each others closest relatives.