The long line is $T_2$, connected (even path connected, also locally 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.

The $\frak{c}$-long line is $T_2$, connected and size continuum $\frak{c}$, but has $2^{\frak{c}}$ many open sets, since there is a size continuum discrete subset.

The more familiar $\omega_1$-long line is $T_2$, connected (even path connected, also locally connected), and size $2^\omega$. There are at least $2^{\omega_1}$ many open sets, since there is a size $\omega_1$ discrete family (such as the centers of the half-open intervals used to construct the long line), and so you can place intervals around each of them as you like, making $2^{\omega_1}$ many distinct open sets.

If $2^\omega<2^{\omega_1}$, for example, if CH holds (but that hypothesis is weaker than CH), then the ordinary long line itself is an example. But in any case, the $\kappa$-long line is an example for every cardinal $\kappa\geq 2^\omega$.

The long line 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.

The long line is $T_2$, connected (even path connected, also locally 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.