Soon after Z had started, DBZ fights had gotten to a point where a character couldn't defeat a stronger opponent by outwitting or outskilling them, but only by becoming linearly more powerful. Therefore, the main characters had to acquire some radical new transformation or move to overpower and defeat the current arc villain, and the next arc villain had to dramatically overpower them yet again in order to be a believable threat.
The big turning point was when mecha-Frieza and his dad came to Earth right before the start of the Androids/Cell saga.
When Goku and Piccolo killed Raditz, it was very clear that Raditz was technically more powerful than both of them and that victory had only been achieved through a combined maneuver that involved a powerful but unreliable technique and Goku's sacrifice. Raditz was defeated, but it was pretty clear that if another villain of similar strength had showed up, the heroes would have been in dire straits. Same for Vegeta: defeating him took Goku going past his limits and breaking his body in the process, a spirit bomb, every surviving fighter tagging in, and Gohan going great ape and falling on him; when he left, it was definitely unsure what would have happened if he had come back for a round two. And when Goku defeated Frieza, we still didn't know if the SUper Saiyan form was even a reliable transformation that could be used at will, or if it would take some extreme events out of Goku's control to trigger it - and without that form, there would have been no way of defeating Frieza.
After that arc though, the excalation of power got out of control, because it became commonplace for the heroes to take a "training break" and suddendly become wildly most powerful than the last supervillain, transformations became as mundane as any other technique, and it all became a process of heroes getting stronger > foes getting stronger > repeat.