Equation of continuity seemingly violated Flow rate is volume of liquid flown per unit time. Suppose I make the time interval so small that it nearly becomes a time instant. When observed at that time instant the volume flown through larger area must be higher than the volume through lower area.

Shouldn't that be against the equation of continuity which says that the flow rate must be constant.
 A: You are looking at the volume only in time, but you are forgetting how volume is calculated in the first place:
$$dV = A \cdot ds$$
where $A$ is the cross-section area and $ds$ is the infinitesimally small tube length for which you are calculating the infinitesimally small volume:
$$ds = v \cdot dt \tag 1$$
where $v$ is the fluid velocity and $dt$ is the infinitesimally small time. The Eq. (1) is key to understanding where your reasoning is wrong - for the same time $dt$ the fluid covers longer distance through smaller cross-section area, which we prove via mass flow.
The infinitesimally small mass flowing into the tube across $A$ in time $dt$ is:
$$dm = \rho \cdot dV = \rho \cdot A \cdot v \cdot dt$$
where $\rho$ is the fluid density which is constant for incompressible fluids.
Since the mass is not stored in the system, for any two points in the tube we have
$$dm_1 = dm_2 \quad \text{or} \quad A_1 v_1 = A_2 v_2$$
Continuity is not violated since water speed at the two points is different. In other words, for infinitesimal small time $dt$ the fluid covers greater distance $ds$ for smaller cross-section area $A$. You have simply neglected the effect $dt$ has on $ds$ and assumed that $ds$ is constant.
