My wife is a Turkish citizen and we are traveling to Bulgaria by air. On the trip home we will have a short 45-minute stopover in Romania.
Does she need to get a transit visa for Romania?
tl;dr: no. (I presume 45 minutes mean a single ticket.)
List of the states whose citizens are required airport transit visas:
let me tell you that just for transit (viza de tranzit) in the airport of Bucharest (Henri Coandă International Airport) like most of the countries, visa is not required.
The transit visa is granted for up to 10 days (when the applicant travels by car or train) or up to 3 days (when the applicant is traveling by plane) .