This is a very populist discourse. Both are important community service areas. The current budget inefficiency and organizational problems of the Diyanet, even too much engagement with the government party(like almost any other state-funded/based instutions), are justified criticisms and should be discussed in detail. The Diyanet has many problems, but comparing it with many other institutions of the country will be more accurate to understand where the problem lies.
But at the same time, it should not be forgotten that Diyanet is an republic instution. Why Atatürk created this institution, what is its purpose of existence, mostly we not think enough about it. (a little reminder, all of the revolts that took place between 1920 and 1930 were about religious fanaticism or bigotry which are all of them out of State control. You can create regional instability with minor religious groups. However, you can destroy the state with the major structures. 2016 was the last example of this.) This is not Norway, you have to act in accordance with the sociological structure of the society and geography. Therefore, the discussion should be based on individuals and their policies, not institutions.
Diyanet should primarily be under the jurisdiction of the Turkish Grand National Assembly. With regard to the budget, the main issue that needs to be examined is the transparency of extra-budgetary income sources, not its budget share from the treasury. Parliament can make this issue more transparent. Another important point is, Whoever pays your salary has authority to supervise you. We go back to a hundred years ago if the dervish lodges and congregations that Atatürk closed down began to pay the salaries of religious personnels. Diyanet is the main factor that provides the difference in influence over between Sunni dynamics and Alevis or non-Muslim communities. For this reason, the Diyanet's budget is a populist area of discussion, but it also constitutes the basic fear of communities outside of this umbrella with its check and balance function.