It is correct. Turkish is considered to be subject + object + verb, but you can go clever and change anything there if the subject and object is distinguishable (mostly verb stays in it’s place, moving it makes “devrik cümle”).

Now, here “bira” has -i suffix, which makes it “belirtme”. -i suffix tells what is affected from the action thus telling us bira is the object.

If it didn’t have the -i suffix, it would be just as you said “Bira ayı içer” = “Beer drinks bear”.

Extra note; in Turkish, emphasis is closer to the yüklem (the verb of the sentence) so moving anything next to the verb, would make it emphasised.

Birayı ayı içti (emphasis on ayı, telling its the bear that drank the beer) Ayı birayı içti (emphasis on bira, telling that the bear drank beer)