I realize errors always occur when I use online software, even for wolfram alpha.
I personally found the results with the first guess of 2 are
$$n_1 = 1.7..., \quad n_2=1.64....$$
and so on, but they are not shown in wolframalpha. Did I go something wrong? or is newton's method just not reliable?
The problem here is not Wolfram but the hand calculation. Starting with $n_0=2$, the correct value for the next step is $$n_1=2-\frac{(2^3-3)\log(2^3-3)}{3\cdot 2^2}\approx 1.3294$$
Alas, the hand method now grinds to a halt, as $n_1^3-3\approx -0.6505$, and you can't take the log of that without complex numbers. Wolfram does indeed do this, but then the method grinds to a halt soon after, as the results spiral out on the complex plane and do not converge.