Frequency of square wave
You're confusing bandwidth with the fundamental frequency, or repetition rate.

A square wave behaves the exact same way as a sine wave, in that as its fundamental frequency increases, you will see more cycles in a given amount of time. 

Square waves theoretically have infinite bandwidth. (I seem to recall seven times the fundamental as a practical rule of thumb from school.) Intuitively, more higher harmonics are needed to sharpen the rising and falling edges.

Plotting it out as a summation of sines is easy and will help with your understanding.