What is the Reynold's number of a plate moving through a fluid?

With the Reynolds Number:

$$text{Re}=frac{rho v L}{mu}$$

the characteristic length $L$ that can be used here is (e.g.):

$$L=frac{l+h}{2}$$

assuming that $L gg w$

Why is the formula for the characteristic length $L$ what you have written it to be and why does $L$ need to be much bigger than $w$?

In many cases, the choice of a 'characteristic length' for the determination of $text{Re}$ can be somewhat arbitrary and there's no 'hard and fast rule' to guide us. The case of a rectangular plate is such a case.

However, the point of transition from fully laminar to fully turbulent flow in terms of $text{Re}$ number is not sharp. Here's Wikipedia's note on this:

For flow in a pipe of diameter $D$, experimental observations show that for "fully developed" flow,[n 2] laminar flow occurs when $text{Re}_D < 2300$ and turbulent flow occurs when $text{Re}_D > 2900$. At the lower end of this range, a continuous turbulent-flow will form, but only at a very long distance from the inlet of the pipe. The flow in between will begin to transition from laminar to turbulent and then back to laminar at irregular intervals, called intermittent flow. This is due to the different speeds and conditions of the fluid in different areas of the pipe's cross-section, depending on other factors such as pipe roughness and flow uniformity. Laminar flow tends to dominate in the fast-moving center of the pipe while slower-moving turbulent flow dominates near the wall. As the Reynolds number increases, the continuous turbulent-flow moves closer to the inlet and the intermittency in between increases, until the flow becomes fully turbulent at $text{Re}_D > 2900$. This result is generalized to non-circular channels using the hydraulic diameter, allowing a transition Reynolds number to be calculated for other shapes of channel.

Because of this uncertainty on $text{Re}$, some uncertainty on $L$ is not problematic. Without direct observation, whether flow will be laminar of turbulent will always be a bit of and 'educated guess'.