According to Wikipedia, the net outflow from the Black Sea is twice the inflow, so a dam would only cause it to rise until it overflows and washes Istanbul away.
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Sorry, but I am having difficulty following your logic.
If you reduce inflow by 1/3, water levels should drop.
And lower water levels mean less outflow because out flowing water has difficulty topping the Dardenelles. The Black Sea would shrink from a Sea to a brackish lake, to salt marsh and might dry up completely like Death Valley or the aural Sea.
As for a dam bursting and flooding Istanbul ..... I thought one of the goals was to generate hydro-electric power, which requires carefully controlled outflow gates directing water to generator turbines???????