Japanese verbs as a class
The Word class system section of the Japanese grammar Wikipedia article does currently state that Japanese verbs and adjectives are closed classes. Strictly speaking, I don't think this is quite correct -- closed classes don't have new members, and generally don't have many members. Japanese verbs are quite numerous indeed, and it's not uncommon for new ones to pop up -- witness recent neologisms like ググる "to Google", or スタバる "to go to Starbucks". Sure, these are slang, but they're also new verbs. It's also not unheard of for folks to coin new verbs by just tacking a -る on the end, such as クッキングる "to cooking" > humorous for "to cook", or ドライビングる "to driving" > humorous for "to drive [a vehicle]".
That said, I think it would be accurate to say that -jiru verbs are a closed class. These all seem to have arisen through regular sound changes from [single kanji]
+ transitive verb suffix す.
Shinjiru and other -jiru verbs
As an addendum to Dono's answer, Shogakukan's entries suggest that these [single kanji]
+ じる terms were originally used as verbs in Japanese by simply appending す, the transitive auxiliary verb (superseded by modern する). This す then became ず due to rendaku caused by the final ん in 信{しん}, and then became ずる when the attributive and terminal forms for verbs fused, and finally became じる due to the shift to ichidan verbs that Dono mentions.
You can track some of this back via online resources by starting at the Daijirin entry for 信じる. Note how it says:
- 「 信ずる 」に同じ。
Same as shinzuru.
Click on 信ずる and you'll jump straight to the 信ず entry. The second line here says:
- 「信ず」の口語形としては、サ行変格活用の動詞「信ずる」が対応する。
The colloquial form of shinzu is the sa-hen conjugation verb shinzuru.
So we have a clear historical progression:
[single kanji ending in ん]
+ す →
[single kanji]
+ ず →
[single kanji]
+ ずる →
[single kanji]
+ じる
Other examples include:
案{あん}じる 演{えん}じる 応{おう}じる 禁{きん}じる 準{じゅん}じる
生{しょう}じる 談{だん}じる 転{てん}じる 封{ふう}じる 論{ろん}じる
Other terms borrowed from Chinese and used as verbs in Japanese historically took the form [kanji term]
+ す, with す again changing to する when the terminal and attributive verb forms merged. Rendaku generally didn't happen for multi-character words ending in ん, which is why you'll only see these -じる forms for single-character on'yomi terms, and mostly (but not always) where the on'yomi ends in ん.