The vast majority of classical Jewish sources, including all the Geonim, and most Rishonim, state that the Midrashim (exegetical rabbinic literature) reflect the views of individual rabbis; not the unanimous consensus of Judaism as a whole. Accordingly, one is free to independently evaluate them.
For example R. Sherira Gaon quoted by the Sefer HaEshkol (Hilkhot Sefer Torah) writes:
ואמר מר [רב] שרירא הני מילי דנפקי מפסוקי ומקרי מדרש ואגדה אומדנא נינהו, ויש מהן שהוא כך...והרבה יש שאינו כן, כגון מה שאמר ר' עקיבא דמקושש היינו צלפחד...והם הזכירו דעתו של כל אחד ואחד, ואנו לפי שכלו יהולל איש. וכן...תנחומא...וזולתם, רובם אינו כן, ולכך אין אנו סומכין על דברי אגדה. והנכון מהם מה שמתחזק מן השכל ומן המקרא מדבריהם,
These words that are derived from verses and are called midrashim or aggada are estimations (umdena)...Some are indeed correct, but many are not correct...And they mentioned the views of each individual, but we follow our own blessed minds... Therefore we do not rely on aggada...Accept as reliable only those that follow from logic or from the verses. (Also cited in the introduction to Menorat Ha-Maor)].
R. Hai Gaon cited there writes:
הגדה ומדרש אף על פי שכתובין בתלמוד אם לא יכוונו ואם ישתבשו אין לסמוך עליהם, כי כללינו הוא אין סומכין על ההגדה
Haggada and Midrash; even if they are written in the Talmud, if they do not make sense disregard them, for we have a principle that we do not rely on the aggadot.
Furthermore, Rav Hai Gaon is quoted (Otsar HaGeonim Berakhot; Peirushim: 67) as saying that not only do the aggadot reflect at most individual views, rather than the views of Judaism, but that even these ideas were often mere suggestions; not definitive ideas.
הוו יודעים כי דברי אגדה לאו שמועה הם, אלא כל אחד דורש מה שעלה על לבו כגון אפשר, ויש לומר, לא דבר חתוך
Know that words of aggada are not based on tradition, rather each person would expound what would enter his mind, such as conjectures, and possibilities; not set things.
For a lengthy survey of classical approaches to the nature and veracity of these sorts of stories found in the Talmud and other works, see here.
Many of these anti-Semitic sites, falsely claim that the Jews agree with everything in the Talmud, that it is their Bible and the like. While that is largely true in the legal realm, in the non-legal realm, that certainly is far from the dominant Jewish view.
Additionally, many commentators primarily the Spanish Rishonim of the 11th-15th centuries, and their successors, frequently suggest that non-literal Midrashim are not to be taken literally.
In this case, for example, the Spanish school explained this non-literally and were horrified by the simple presentation of this Midrash.
R. Isaac Arama, for example, writes in his Akedat Yitshak (Genesis: Sha'ar 8) that certainly the intent is not that he engaged in intercourse with the animals, but rather that he mentally probed them and evaluated them, and found them all lacking:
וזו היא כוונתם ז"ל באמרם שבא אדם על כל בהמה חיה ועוף ולא נתקררה דעתו בהם (יבמות ס"ג א). ירצה שבא בדעתו וטוב התבוננותו עליהם ועל טבעם ולא נתקררה דעתו שיהיה זווגו
And this is their intent in that which thy said that Adam came upon all the animals and wasn't satisfied by them. That is, that he came to them through mental evaluation and deep contemplation of them and their nature, and he was not satisfied any could serve as a match for him.
R. Abraham Saba writes very similarly in his Tseror HaMor to Genesis. As do R. Don Isaac Abravanel in his commentary to Genesis (end of 2:19), and R. Samuel Almosnino (23).[i]
Similarly, R. Isaac Karo writes in his commentary to Genesis (2:23):
חס ושלום שבא בפועל
Heaven fore-fend that he literally engaged in intercourse.
[i] Cited in Prof. Eric Lawee's The Reception of Rashi’s Commentary on
the Torah in Spain: The Case of Adam’s Mating with the Animals, p. 57. In The Jewish Quarterly Review, Vol. 97, No. 1 (Winter 2007).
