"You're" is short for "You Are" ... so "you're welcome" is quite correct "Your" is possessive term ... "it is your bike" ... not "your welcome" .
you are welcome [your'e welcome ]
you're welcome, the apostrophe is an a, which is you are welcome, your is a possessive pronoun and shows somebody owns something, like this is your car
You're welcome
you're welcome, because your refers to possesion. you're is short for you are. you are welcome. You are welcome--You're welcome.