“If evolution is not true, then why does the Bible say that animals and birds came out of the water (Genesis 1:20-21)?”

It doesn't say that.  The Bible teaches that birds were formed from the ground.  Let’s read the passage in question from the King James: “And God said, ‘Let the waters bring forth abundantly the moving creature that hath life, and fowl [that] may fly above the earth in the open firmament of heaven.’ And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that [it was] good.”

Notice those words in brackets? They are not there in the Hebrew and have been inserted for easier reading in English. What does it say in the Hebrew? The poetry alone is quite impressive:

“And God said, ‘Let the waters teem/swarm [“sharats”] with teeming/swarming things [“sherets”] that have life, and flying creatures [“owph”] flying [“uwph”] in the open expanse of heaven. And God created great sea monsters/dragons/whales [“tanniyn”] and every living soul that moves which teem/swarm the waters after their kind, and every flying creature after his kind: and God saw that good.”

Immediately one cannot help but notice a slight difference in the description, the most obvious difference being the archaic language of the King James and the even more archaic language of the Hebrew. We also must recall that we approach literature differently than the culture to which this was initially written.  The problem concept here is "bring forth."  Do we today have a 1611 understanding of the term?  Hardly.  Besides if at creation birds were "brought forth" from the sea, then what do we do with Noah who must have been painfully pregnant in Genesis 8:17 when God said, "Bring forth with thee every living thing that [is] with thee, of all flesh, [both] of fowl, and of cattle, and of every creeping thing that creepeth upon the earth; that they may breed abundantly in the earth, and be fruitful, and multiply upon the earth."

They key question to be answered is this: is the purpose of the text to describe the detail of the creation of animals; or, does it describe their habitation? The latter is more reasonable because the purpose of this particular passage is not detail of “how,” but “what:” Teeming/swarming things live in the sea and flying creatures inhabit in the sky.

So if they did not come from water, where do the birds/flying creatures come from? Once the reader understands the general story in Chapter 1 (that all things are created by and are distinct from The Creator) the author can lift out one detail and expound on it. In this case, we can focus on these particular creatures. Interjected into the details of the creation of man, who incidentally was formed out of the ground (Genesis 2:1), we find this statement, “And out of the ground the LORD God formed every beast of the field, and every fowl of the air.” (Genesis 2:19)