Whenever science and the Bible contradict each other, why do some people naturally assume that the Bible is wrong?

Science (correctly understood) and the Bible (correctly understood) never contradict each other. Truth cannot contradict truth. Which is why the teaching of the original Christian Church, to which Jesus Christ promised the fullness of God's truth, never conflicts with natural truths revealed by science. Manmade denominational churches that have defected from God's Church and attempt to find truth in "the Bible alone" have many false beliefs, as shown by the fact that the teaching of each denomination contradicts the teachings of other denominations, and truth cannot contradict truth. So, it is no surprise that some of these false teachings also conflict with natural truths clearly revealed by science.

Science is supported by evidence while the Bible is just ancient mythology.  Pretty simple.

There's very little proof on anything in the bible, science is based on proof. Thats why

It's usually the other way around.  Some people (fundamentalists) insist that the Bible is 100% correct and then they have to do mental gymnastics to fit that to what science has found.

The average person in the pews is usually just parroting what the professional apologists have come up with.

Judges 19:26-29 – “At daybreak the woman went back to the house where her master was staying, fell down at the door and lay there until daylight. When her master got up in the morning and opened the door of the house and stepped out to continue on his way, there lay his concubine, fallen in the doorway of the house, with her hands on the threshold.

He said to her, "Get up; let's go." But there was no answer.

Then the man put her on his donkey and set out for home.

When he reached home, he took a knife and cut up his concubine,

limb by limb, into twelve parts and sent them into all the areas of Israel.”  

The answer to your question: Jesus only inhabits 

about 15% to 20% of the Bible, which means that 80% to 85% of its texts deal with life as it was lived 2,000 years ago. Even when there is no need to apply Science, read the Bible as it is written and try to imagine why God would inspire any kind of texts like the one above which tend to dominate that all-important and remaining 80% to 85% of the Bible where Jesus doesn't even come into play. 

When it is science, that is constantly proving itself wrong

Pride, is the answer....thinking you know it all

You will have to give an example. I am unaware of anywhere that science and the Bible contradict.