Difference between two derivations of length contraction?

I’m working through SR along with a lecture/class series. Last week, we worked through an elementary proof of length contraction after already deriving that $t = gamma t_0$ with $gamma$ the familiar Lorentz factor.

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This is the proof in the lecture notes:

Consider an observer B and a mirror. They are at rest with respect to each other, separated by a distance $L_0$. The observer reflects a beam of light onto the mirror to measure its distance.

$$L_0 = frac{ct_0}{2}$$

Now consider an observer A. Observer B is moving away with velocity $v$ with respect to A. A also measures the length with B’s experiment:

$$L = frac{c}{2} (Delta t_1 + Delta t_2)$$

Where the terms in brackets represent the motion of the light to the mirror, and the motion of the light from the mirror back to B.

$$Delta t_1 = frac{L}{c} + frac{Delta t_1 v}{c} text{ and } Delta t_1 = frac{L}{c – v}$$

Analogously follows $Delta t_2 = frac{L}{c+v}$.

We now obtain:

$$Delta t_1 + Delta t_2 = frac{2L/c}{1-v^2/c^2}$$

Using that $Delta t_1 + Delta t_2 = t_A = gamma t_0$ we derive

$$gamma t_0 = frac{2L/c}{1-v^2/c^2}$$

$$t_0 = gamma frac{2L}{c}$$
$$frac{ct_0}{2} = gamma L$$
$$L_0 = gamma L$$

Which is the familiar equation for length contraction.

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Here my issue starts.

I was recreating the derivation when I figured "hey, why not make the substitution for gamma earlier." So, I took

$$Delta t_1 + Delta t_2 = t_A = gamma t_0$$

And substituted it into

$$L = frac{c}{2} (Delta t_1 + Delta t_2)$$

This gives

$$L = frac{c}{2} gamma t_0$$
$$L = frac{ct_0}{2}gamma$$
$$L = L_0 gamma$$

The reverse of the equation I derived and should have derived. Thus, there must be an illegal operation somewhere. But where?