Q.What type of bush is this?
This bush is crazy. If we cut it down to an acceptable height for a bush, 2 weeks later is even bigger than before we touched it. Our landlords came and cut it down to a mere stump once. And now it’s even more out of control! I’ve tried looking into what this thing could be, and all I can tell from my research it’s Mulberry… but there is no fruit on the thing… Any answers would be wonderful! Thank you!

I believe this to be a Viburnum. The exact cultivator or variety is unknown.
With all the pruning you are likely affecting its blooming period.
https://www.gardeningknowhow.com/ornamental/shrubs/viburnum/viburnum-shrub.htm