Are conservatives afraid of transgender people becoming more socially accepted?

Not at all the terrifying aspect is coming up with new laws to force this on people. 

They sure seem to be. Remember though, most conservatives believe that humans come from two people in a magic garden, and that the first was created from dirt. 

No

They feel like women's sports should be reserved for women born as girls and vice versa regarding men's sports.

Also, we need to reject those who refuse to identify as a man or a woman. If you're opting to identify as a bisexual penguin or a lesbian stop sign, or even an alien or a cyborg from the future, you need mental help ASAP.

No, because it isn't gonna happen

Gee, what IS the latest definition of someone that is "transgendered"?

How about a "MENTAL ILLNESS"?

jEWDY AND Charlene had their sons testicles remove at birth!

Thanks for an honestly good question. 

CONServatives are REALLY AFRAID FOR THEIR CHILDREN to see these people, get to know them socially and to accept them as human beings.  Why?  Because if their own child is trans (from birth) then the parents will have a hard time controlling them from being themselves and living their own lives.

It's control.  CONS control other people using hatred and fear.

There is no such thing as "transgender people."

No amount of pseudo-medical mutilation and quackery can transform a man into a woman or vice versa.

You should be ashamed of yourself.