Synchronisms between the reigns of the northern and southern kingdoms (Israel and Judah) are all affected by an apparent discrepancy of a few years. The case of Joash’s 23rd year is particularly affected, but the basic reason is the same as other kings. It relates to whether a kingdom uses the 'accessional' or 'non-accessional' method of dating.
In both cases the part-year beginning and ending a king’s reign are counted as two of his years, even though they were less than twelve months, but Judah treated the first full year as 'number 1.' They had the 'accessional' system of dating where the incomplete portion of months from a father's death to the end of the regnal year became the sons 'accession' - not his 'first year.'
However, the non-accessional system calls the first portion of a kings reign his 'first' year. This was the system adopted by Jeroboam when the northern tribes broke away from the house of David. (These matters came to light mid-last century with the chronological research done by Dr. Edwin Thiele)
Now, getting back to Joash. Shortly before him, the throne of David had been usurped by Queen Athaliah, daughter of Ahab, king of Israel. She immediately began to destroy the house of David, and impose foreign practises. Although not explicitly stated, Judah’s accessional system is likely to have been tampered with in order to obscure the lineage of David. Hence, Joash would have begun his reign with the system Athaliah had installed, continuing as a ‘hangover’ from her changes.
Therefore, it is my thought that some of Joash’s reign was reckoned using a non-accessional count, his accession counting as 'number 1' and his first full year as 'number 2' and so on. If we make this assumption, the 23rd year of Joash works very well! However, by the time of his 37th year, the accessional system had been re-established. It probably was rectified later in the reign of Joash.
Please examine the diagram here. (scroll down to Jehu and Joash)