You *DO* put hurt in past tense. it is just irregular, and all 3 tenses look the same. The past tense of "hurt" is "hurt", just like the past tense of "eat" is "ate", not "eated." Another verb where the past is the same as the present is "put".
I injure myself every day= I hurt myself every day.
I injured myself yesterday = I hurt myself yesterday.
I have injured myself many times= I have hurt myself many times.
present /past/past participle
injure /injured/ injured
eat/ ate/ eaten
hurt/ hurt/ hurt
put/ put/ put