In Sweden, bow hunting is illegal.......

Hunting with airguns is resticted to small game, up to magpie.

However, big bore airguns are classed the same as 22LR if they have the same power (150J/115fpe).

So, I guess that any game that You would hunt with a 22LR is fair game for a big bore airgun?

We don't have any (wild) turkeys in Sweden, but birds of similar size reguire a 22WMR and deer require a centre fire rifle (222Rem).