If you took any Indo-European language and created a new language completely contradicting it with respect to any aspect of the language, you'd get something pretty close to Turkish :-) I am really surprised you think Turkish is similar to any Indo-European language. Btw, the Altaic language family is also disputed. The most accepted theory is that Turkic languages are an isolated language family by themselves.

I am sure I had learned (long in the past) that they were in different families; but my little exposure to Turkish led me to believe (having forgotten what I had learned) that they were related; why? because the conjugations and tense modalities evoked to me an analogy to those of romance languages, and the declination cases and flexible word order evoked to me those of Easter European languages; while at the same time dismissing the greatest difference: it's agglutinative character.

That’s correct, grammar wise Turkish is closer to Japanese and Korean than to Indo-European languages, but there’s an incredible amount of French loanwords (partly because of Ataturk’s obsession with French).

French loanwords started came to language in first half of 19th century, during the attempts of reforms by Sultans and with the influence of people who had French education. I would argue most french loanwords were adapted earlier than 1920's where Ataturk had an influence on Turkish language. Roughly ~4% of Turkish words are French loanwords.

Turkish has little structural or grammatical resemblance to Indo-European languages from the perspective of its main features (extreme agglutination and vowel harmony).

(Edit: reduced absoluteness, because lack of grammatical gender feature can be found in some Indo-European languages)