부끄려요 - (present tense of the verb 부끄리다) feel ashamed of, be embarrassed by
부끄러워요- (from the adjective 부끄럽다) shameful, embarrassing etc. /that makes one feel ashamed or embarrassed.
내 실수가 부끄러워요 should really be translated as:
My mistake is embarrassing.
despite the interpretation being:
I feel embarrassed by my mistake.
The Naver Dictionary defines the Korean word for ashamed, shy, abashed, etc. as 부끄럽다 .
Korean predicate (verbs and adjectives) are either regular or irregular verbs. Predicate that do not undergo a change of their respective stem based on the sound of the added suffix are irregular verbs. However, those predicates that undergo a change in their respective stem based on the added suffix are irregular verbs. ㅂ irregular verbs are subject to the following conjugation. When the suffix begins with a vowel ㅂis deleted and either 우 or 오 is added. Adding either 우 or오 follows standard rules. The verb in this sentence is 부꾸릅다. ㅂ is deleted and 우 is added changing the predicate to 부끄러우. After this the suffix that began with a vowel is added resulting in 부끄러워요.
부끄럽다 부끄럽스니다 부끄러워요. Source: Basic Korean, A Grammar and Workbook by Andrew Sangpil Byon, Published by Routlege 2009 (pages 151 – 157. The procedure is also detailed Korean a Comprehensive Grammar bu Jaehoon Yeon and Lucien Brown, published by Routlege 2011.