The sequence $2023$ appears $13$ times in the writing of $2023$, $12023$, $22023$ and of $20230$, $20231$, $20232$, $\ldots$, $20239$. Moreover, $2023$ can appear by connecting the end of a number to the beginning of the next number. The writing $202|3$

appears only once, in $3202|3203$, and the writing $20|23$ appears $11$ times, in $2320|2321$ and from $23020|23021$ to $23920|23921$. The writing $2|023$ is impossible, because no positive integer begins with $0$, therefore the number of occurrences of $2023$ is $13 + 11 + 1 = 25$.