The [long line](https://en.wikipedia.org/wiki/Long_line_(topology)) is $T_2$, connected (even path connected), and size $2^\omega$. 

But there are at least $2^{\omega_1}$ many open sets, since there is a size $\omega_1$ discrete family, and you can place intervals around each of them as you like, making $2^{\omega_1}$ many distinct open sets.

If $2^\omega<2^{\omega_1}$, this gives an example. This hypothesis holds under CH, but it is weaker than CH.

In general, if you take the $\kappa$-long line, for uncountable cardinal $\kappa$, then it is $T_2$, connected and size $\kappa\cdot 2^\omega$, but has at least $2^\kappa$ many open sets. So if you take $\kappa\geq 2^\omega$, it will give a counterexample without any CH assumption.