How can I explain the error of pronunciation in the sentence
I sink I'm going to bed
where the word pronounced sink is actually think?
Technically, there are two ways of pronouncing -th correctly. The voiced dental fricative /ð/ as in this and mother, and the voiceless dental fricative /θ/ as in thing and thin. But many teachers will simply say voiced and unvoiced.
The -th in think is unvoiced, meaning, only air passes through the mouth
Wikipedia explains what the OP refers to "error of pronunciation" can be called
th-alveolarization In rarer or older varieties of African American Vernacular English, /θ/ may be pronounced [s], as in bathroom [ˈbæsɹum].
Children with a lisp, however, have trouble distinguishing /θ/ and /ð/ from /s/ and /z/ respectively in speech, using a single /θ/ or /ð/ pronunciation for both, and may never master the correct sounds without speech therapy. The lisp is a common speech impediment in English. Foreign learners may have parallel problems. In English popular culture the substitution of /z/ for /ð/ is a common way of parodying a French accent, but in fact learners from very many cultural backgrounds have difficulties with English dental fricatives, usually caused by interference with either sibilants or stops. Words with a dental fricative adjacent to an alveolar sibilant, such as clothes, truths, fifths, sixths, anesthetic, etc., are commonly very difficult for foreign learners to pronounce.
For further information please see this question on EL&U
Is there a rule for pronouncing “th” at the beginning of a word?