Where in the Bible does it say we have a sin nature? Romans 5:12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:

Notice the verse says death is passed on and not that sin nature is passed on to all men. Many denominations make our sin nature as part of their doctrinal statement. 

searching under doctrinal statements and checking some links:

Southern Evangelical: ( see http://ses.edu/about-us/doctrinal-statement ) says in point 9 "total depravity of human nature, has been transmitted to the entire human race"

Ariel Ministries: ( http://ariel.org/doctrinal-statement.htm ) says under Man: Created and Fallen "totally depraved; that this fallen nature is transmitted to every descendent of Adam"

We see that Romans 5:12 says "death" and the statements say "depraved" or "depravity" and I have heard the term "sin nature" used by other organizations. Are these doctrinal statements twisting scripture?

In fact in Calvinism we have TULIP where the T stands for Total Depravity so perhaps this doctrine is Calvinistic in nature (pun intended).
The phrase " sin nature " is not used in the bible I use, KJV. However, there are verses that imply such.

Psalms 51;5 
 Behold,I was shapen in iniquity,and in sin did my mother conceive me.

Psalms 58;3
The wicked are estranged from the womb:they go astray as soon as they be born, speaking lies.

Ephesians 2;3
Among whom also we all had our conversation in times past in the lusts of our flesh, fulfilling the desires of the flesh and of the mind; and were by nature the children of wrath, even as others.

We are all children of wrath until we are born again, John 3;5.