Checking with Google Translate, using languages I know sufficiently well to check the result:
"I saw John coming with a ball, but Kate saw John throw it"
Turkish
John'un topla geldiğini gördüm ama Kate John'un topu attığını gördü
John-Gen ball-INSTR coming-POSS.3Sg-ACC see-1Sg.PAST but Kate John-GEN ball-Acc throwing-POSS.3Sg-ACC see-3SG.PAST
I have seen -dik called an "Action noun" in descriptions of Turkish. The genitive plus possessive plus case marking clearly shows that it's a nominal construction anyway. GT repeats "the ball" instead of using a personal pronoun, but I think onu "it-ACC" could have been used as well.
Kazakh
Мен Джонның доппен келе жатқанын көрдім, бірақ Кейт Джонның оны лақтырғанын көрді
Men Djonnıñ doppen kele jatqanın kördim, biraq Keyt Djonnıñ onı laqtırğanın kördi
I John-Gen ball-INSTR come-GER lie-GER-Poss.3Sg-ACC see-1Sg.PAST but Kate John-GEN it-Acc throw-GER-POSS.3Sg-ACC see-3SG.PAST
A construction using gerunds; jatyr "lie", like "stand" and "sit", can be used to form the progressive with the present gerund of the verb (as here in kele jat-). Here as well, the genitive plus possessive plus case marking clearly shows that its a nominal construction; you can use nominative subjects and gerunds without possessive and case marking only when the subject of the dependent and the main clause are identical.
Uzbek
Men Jonning to'p bilan kelayotganini ko'rdim, lekin Kate John uni tashlaganini ko'rdi
I John-Gen ball with come-PROGR-GER-POSS.3Sg-ACC see-1Sg.PAST but Kate John it-ACC throw-GER-POSS.3Sg-ACC see-3SG.PAST
Basically the same constuction using gerunds as Kazakh has, only the Kazakh auxiliary in the first clause has become a suffix in Uzbek. I would have expected John to be in the genitive in the second clause as well, don't know whether the NOM is a usage I don't know or a mistake by GT.