lower bound of majority function?
You haven't specified exactly what kind of circuits, but in general since parity $\oplus$ is in $\mathsf{TC^0}$ and majority $Maj$ is complete for $\mathsf{TC^0}$, you get a similar subexponential size lower-bound for majority. 

A simple reduction would go through counting function $NumOnes$ that counts the number 1s in the input in binary. This would give $2^{n^{\Omega(1/d)}}$ where constant in the exponent comes from the size of $\mathsf{AC^0}[Maj]$ circuit computing $numones$. I don't know the best value of the constant but it shouldn't be big.

On the other hand, there are $2^{n^{O(1/d)}}$ circuits for majority as majority is in $\mathsf{NC^1}$ and any $\mathsf{NC^1}$ circuits can be converted into a subexponential-size bounded-depth $\mathsf{AC^0}$ circuit (by performing a layered brute-force).