Having observed this map on the front page, I noticed that Poland, Turkey, and Vatican City all legalized homosexual relationships well before countries like Sweden, Norway, Finland, Britain, etc, countries that I would have thought to be the first to legalize homosexual relationships. How and why did these countries that are mostly conservative Catholics or conservative Muslims beat countries that pride themselves on their progressiveness?
This largely boils down to the synchronic vs. diachronic thinking. The former is generally described as the 'point in time perspective' while the latter is, conversely, 'through-time perspective'. In other words, synchronic attribution of some traits to a specific subject is based on a highly specific and easily identifiable chronological state (e.g. how does the country look in 20th century) and then projected to other periods. Diachronic thinking is basically the opposite, as it also takes into account changes that have occurred at various points in time and thus is less prone to various biases. I suspect that the author of the question is aware of this distinction and thus have put the designators like 'socially conservative' in parentheses.
Now, let's put the above explanation in practical perspective. The designators like 'socially conservative' or 'progressive', when attributed to the countries or regions usually allude to a very specific state of affairs and in this case it a very recent history, hailing back to no more than 50 years, i.e. encompassing late 20th century and the first two decades of 21st century. This is fraught with intrinsic error wen speaking of history, as the countries, people, cultures and communities invariably change through their existence, sometimes quite dramatically. It should also be noted that the 'progressive' tendencies are caused precisely because of their lack in previous times in an attempt to 'make up' or 'atone' and thus can be more fervent than in the countries that never experienced such problems or solved them long ago. A good example is harsh treatment of the display of Nazi-related symbols and content in Germany or the anti-racist policies in USA that are far less pronounced in countries that do not have history of genocide or slavery and racial segregation respectively.
Now, the case of Poland is quite curious, because technically, it is one of the few countries that have never, as a state, made the homosexual acts a crime. The year 1932 alludes to the adoption of the new penal codex that replaced the legislative acts introduced by the countries governing Polish lands after the partitions of 1772, 1793 and 1795 that eventually resulted in the dissolution of the Polish-Lithuanian state and complete annexation of its territory by three neighbouring countries. Even though Poland regained independence in 1918, legislative process took a lot of time due to the highest priority being assigned to the creation of the basic, unified framework for the political and legal systems and the political turmoil in the early 1920s that resulted in frequent government changes (if memory serves, the chamber of ministers changed 11 times within 6 years at that time) what slowed down all the legislative processes. Nevertheless, until 1932, Poland used three separate penal codes: Russian Penal Code of 1903 in the area formerly annexed by Russia, Austrian Penal Code of 1852 in Austrian partition and German penal code of 1871 with further additions in the lands claimed by Prussia. To create a unified Polish legislation, in 1919 Polish Parliament created a Codificative Commission composed of the renowned lawyers who were assisted by various specialists in the fields of forensics, economy, psychology etc. They came with the new unified codex in mid-1931 that has been approved by the parliament and president in the subsequent year. As to why homosexual acts were excluded from the 1932 Polish penal code, there were several arguments for this development. Primarily, as this was a brand-new code, acts that were not considered socially harmful could have been simply not included as in legislative practice it is easier to omit specific regulations from new code that to remove an existing law (there are many reasons for this, from lac of interests in non-crucial laws to avoidance of antagonization of some part of potential voters). Additionally, Poland had no history of penalization for this particular act.
Now, even though Poland is considered a conservative Catholic country today, although the veracity of this statement is up for debate, such stance is largely a product of the Communist era. Territorial cessions in the east precipitated by the Tehran and Yalta conference as well as the annihilation of the majority of Jewish population that in 1939 reached 13% overall and up to 45% in several cities meant that Poland in 1945 became a national and religious monolith, having lost their Ruthenian, German, Orthodox and Jewish population that became reduced to small minorities. Furthermore, as communist government was more or less hostile towards the Catholic Church, this made the latter a natural ally of the democratic, anti-communist underground that gained political prominence since 1989 (and later gained political prominence due to low popularity of the left-wing politicians largely seen as the supporters of the former regime). Last but not least, the position of Church in Poland was additionally elevated after the election of Polish cardinal Karol Wojtyła as the new pope John Paul II in 1978.
This said, it should be noted that Poland and Polish-Lithuanian Commonwealth were, throughout late Middle Ages and Early Modern period far more 'progressive' and 'liberal' (using modern perception of these terms) due to significant ethnic and religious diversity that necessitated compromises and coexistence. When the Western Europe experienced political turmoil and religious wars largely resulting from the Reformation and coming to an uneasy truce with the Peace of Augsburg in 1648, Polish-Lithuanian Commonwealth enjoyed the equality of all Christian religions and tolerance for non-Christian faiths (mainly Judaism and Islam) codified in Warsaw Confederation Act of 1573 and resulting from centuries of peaceful coexistence between people of different faiths since the union between predominantly Catholic Poland and largely Orthodox Lithuania with Jews, massively arriving to Poland in the wake of persecutions in the West since mid-13th century or Muslim Tatars settling n Poland since very late 14th century being prominent minorities.
As to why homosexual acts were not penalized in Poland and Polish-Lithuanian Commonwealth, historians are not entirely sure, but it is generally derived from the relatively early religious tolerance mentioned above and the specific development of the 'noble democracy' heavily inspired by the ancient Athenian government and thus took the form of class oligarchy, with the relatively numerous (10-12% of the total population) and legally uniform nobility (meaning that they had equal rights regardless of their position and wealth) being the most privileged class. These developments, largely caused by the precarious position of Polish monarchs between late 14th and early 16th century who granted various privileges to the members knightly class in an attempt to secure their loyalty (what caused the essentially feudal relationships to remain in Poland for a long time and contributed to the re-emergence of widespread serfdom, but this is another matter). Increased importance and independence of nobility that became de facto leading force in the state politics, compounded with the aforementioned religious tolerance caused the rules promoted by ecclesiastical authorities (such as 'actions against nature' as zoophilia or homosexuality were classified) to lack traction in Poland as nobles treated them largely as an attempt to curb their freedoms.
Conversely, the most numerous peasant class lacked any political influence, representation or self-organization, but at the same time nobility, interested primarily in the economics of their estate, were not interfering much in the private life of their subjects. Additionally, as the position of Orthodox and Catholic Church as an institution was not as strong as in e.g. Spain and Protestantism, although relatively common among nobility and burghers was very rare among peasants, it was not uncommon the pre-Christian practices being followed more or less overtly along the nominal adherence to Christian customs. In such milieu, the potential homosexual acts were more a private matter that could have led to ridicule or ostracism among neighbours but were not a reason for an official trial. As I wrote above, peasants themselves had no possibility of holding any court and the nobility had no interest in punishing their workers for something that did not influence their lives and interests.
I can't speak for anything else, but a few years ago /u/PaxOttomanica gave this fascinating answer about the decriminalisation of homosexuality in the Ottoman Empire (which formerly ruled Turkey and is where the date comes from) https://www.reddit.com/r/AskHistorians/comments/4r17fc/comment/d4xwz1l