I think you've got it a bit mixed up. You are probably thinking of the auxiliary conjugation that is させる (Causative). Тhis is added to a verb to mean either that someone gives permission or that someone forces you to do it. The verb 見せる is indeed related to 見る, but it's a separate verb and not a conjugation of みる. 見れる and 聞こえる mean "are able to see/hear", but are also not conjugations of 見る and 聞く. They only act like a potential form for those verbs.
All verbs can be conjugated into either a potential form or a causative form:
る-Verbs: Remove る and add られる or させる
食べる (to eat) 食べられる (can eat) 食べさせる (allowed to eat/forced to eat)
借りる (to borrow) 借りられる (can borrow) 借りさせる (allowed to borrow/forced to borrow)
う-Verbs: Change the syllable into the え (potential) or あ (causative) in that consonant's row
飲む (to drink) 飲める (can drink) 飲ませる (allowed to drink/forced to drink)
働く (to work) 働ける (can work) 働かせる (allowed to work/forced to work)