When I play Time Warp during a turn, does my turn end immediately, or can I do something afterwards and then get my next turn?

No — playing Time Warp doesn't end your turn. It just inserts an Additional Turn for you into the turn queue right after this one (Rule 735), so you keep playing your current turn normally: finish your Main Phase, take any remaining actions, and end the turn whenever you're done. Then you take that extra turn, and only after it's finished does play return to the normal turn order (Rule 737).

One thing worth flagging: "after this one" means immediately after your current turn — not "instead of my next regular turn." In a two-player game that reads as you taking your current turn, then your extra turn, then your opponent's turn (FAQ 8241), so the extra turn happening first is what denies your opponent their Beginning Phase holding points for that cycle.