Can I use a Planeswalker's ability when I took over its control this turn?
You may use one of the planeswalker's abilities during your turn.

In Bolas&#39;s Clutches does indeed allow you to control planeswalkers, and each player's turn counts separately. If your opponent ended their turn, your turn is separate from that, those two turns do not count as the same turn.


  606.3. A player may activate a loyalty ability of a permanent they control any time they have priority and the stack is empty during a main phase of their turn, but only if no player has previously activated a loyalty ability of that permanent that turn.


Please note that the game structure doesn't have a concept of "rounds". In order of decreasing scope, the structure of MtG is as follows:


Tournament (a series of matches)
Match (series of games)
Game (series of turns)
Turn (series of phases)
Phase (series of steps)
Step (some phases don't have steps)