"I am very happy," Bitshiabu said after touching down in Istanbul. "I hope we can achieve very big things together here.
"I am looking forward to meeting the Galatasaray family and the fans."
RB Leipzig will receive around €2 million for the loan itself, with Galatasaray holding an option to sign the 21-year-old centre-back permanently next summer for a fee in the region of €28 million.
The Turkish champions have also agreed to pay Bitshiabu a fixed net salary of €2 million for the remainder of this season, according to the deal's terms.
The Frenchman joined Leipzig from Paris Saint-Germain's academy in the summer of 2023, initially signing a five-year contract after the Bundesliga club paid a reported €15 million plus bonuses for his signature.
He has made 48 competitive appearances for Leipzig in total, 28 of them as a starter, but game time became increasingly scarce last season, restricting him to just 14 outings across all competitions.
Standing at 1.96 metres, Bitshiabu had been tracked by several of Europe's leading clubs earlier in his career, and RC Lens were also in talks over a potential loan before Leipzig turned instead to Galatasaray in the closing stages of the window. The move reunites him with regular European football, with Galatasaray set to compete in the Champions League this season while also defending their Super Lig title.
Bitshiabu has represented France at Under-17, Under-18, Under-19 and Under-20 level, and was part of the squad that won the 2022 European Under-17 Championship. Born in Villeneuve-Saint-Georges and of Congolese descent, he progressed through Paris Saint-Germain's academy before making his senior Ligue 1 debut as a teenager.
Leipzig extended his contract through to June 2029 last year and removed a €40 million release clause that had originally been written into his deal, a move that reflected the club's continued belief in his long-term potential even as first-team opportunities dried up under increasing competition at centre-back.
An injury-disrupted start to his time in Germany meant he had to fight his way into contention, with his most productive campaign coming two seasons ago when he made 21 Bundesliga appearances, starting 13 of them.
The loan gives Bitshiabu the platform to rebuild momentum in a league where he will be expected to feature far more regularly, with Galatasaray manager Okan Buruk having pushed to add defensive reinforcements before Turkey's transfer window closed on Friday.