It seems to me that the most appropriate simple model is that there are two kinds of people: people who will and won't purchase, given that they visit. The question is what proportion of people (we're assuming an infinite population) belong to each category. That entails a binomial model. Let the proportion of the population who would purchase be $p$, the number of people who visited be $N$, and the number who bought be $R$. Then the posterior inference for $p$ is
\begin{align}
\mathrm{prob}(p | R, N, \mathcal{I}) &= \frac{\mathrm{prob}(R | p, N, \mathcal{I}) \: \mathrm{prob}(p | N, \mathcal{I})}{\mathrm{prob}(R | N, \mathcal{I})} \\
&= \frac{\mathrm{prob}(R | p, N, \mathcal{I}) \: \mathrm{prob}(p | N, \mathcal{I})}{\int_{p'} \mathrm{prob}(R | p', N, \mathcal{I}) \: \mathrm{prob}(p' | N, \mathcal{I}) \: dp'}
\end{align}
Assigning a uniform distribution to the prior, $\mathrm{prob}(p | N, \mathcal{I}) = 1$, and the binomial distribution to the likelihood, $\mathrm{prob}(R | p, N, \mathcal{I}) = {N \choose R} p^R (1 - p)^{N - R}$, we have
\begin{align}
\mathrm{prob}(p | R, N, \mathcal{I}) &= \frac{{N \choose R} p^R (1 - p)^{N - R}}{\int_{p'} {N \choose R} p'^R (1 - p')^{N - R} \: dp'} \\
&= (N + 1) {N \choose R} p^R (1 - p)^{N - R}
\end{align}
For $N = 1382$ and $R = 23$, this posterior looks like this:
This represents how much you now know about $p$, the proportion of the population who would buy the product, if they visited the site.
Your prediction of how many customers would buy it after visiting, given a new number of visitors $n$ and retaining what you now know about $p$ via $N$ and $R$, is another matter.
The distribution for the new number of buyers $r$ is
\begin{align}
\mathrm{prob}(r | n, R, N, \mathcal{I}) &= \int_p \mathrm{prob}(r, p | n, R, N, \mathcal{I}) \: dp \\
&= \int_p \mathrm{prob}(r | p, n, R, N, \mathcal{I}) \: \mathrm{prob}(p | n, R, N, \mathcal{I}) \: dp \\
&= \int_p \mathrm{prob}(r | p, n, \mathcal{I}) \: \mathrm{prob}(p | R, N, \mathcal{I}) \: dp \\
&= \int_p {n \choose r} p^r (1 - p)^{n - r} \: (N + 1) {N \choose R} p^R (1 - p)^{N - R} \: dp \\
&= \frac{(N + 1) {N \choose R}{n \choose r}}{(N + n + 1) {N + n \choose R + r}}
\end{align}
Retaining $N = 1382$ and $R = 23$, and choosing a new visitor sample size of $n = 1000$, the prediction for the number of buying visitors is shown below:
Note that the prediction curve is broader than the curve you'd get if you simply scaled the inference curve from $[0, 1]$ to $[0, 1000]$. This is because it incorporates the uncertainty in the number of buying visitors given a particular $p$, as well as the uncertainty in $p$.
