Generalized Turán problems investigate the maximization of the number of certain structures (typically edges) under some constraints in a graph. We study a game version of these problems, the Constructor-Blocker game. We mainly focus on the case where Constructor tries to maximize the number of triangles in her graph, while forbidding her to claim short paths or cycles. We also study a variant of this game, where we impose some planarity constraints on Constructor instead of forbidding certain subgraphs. For all games studied, we obtain (precise) asymptotics or upper and lower bounds.