Crucifixion was used by Romans for punishing non-Romans from the end of the first century BC. By the time of Jesus, the regulations governing a crucifixion had become relatively standard in the Roman world. The following description is taken from Tzaferis (1985).
Once a defendant was found guilty and was condemned to be crucified, the execution was supervised by an official known as the Carnifix Serarum. From the tribunal hall, the victim was taken outside, stripped, bound to a column and scourged. The scourging was done with either a stick or a flagellum, a Roman instrument with a short handle to which several long, thick thongs had been attached. On the ends of the leather thongs were lead or bone tips. Although the number of strokes imposed was not fixed, care was taken not to kill the victim. Following the beating, the horizontal beam was placed upon the condemned man’s shoulders, and he began the long, grueling march to the execution site, usually outside the city walls. A soldier at the head of the procession carried the titulus, an inscription written on wood, which stated the defendant’s name and the crime for which he had been condemned. Later, this titulus was fastened to the victim’s cross. When the procession arrived at the execution site, a vertical stake was fixed into the ground. … If the victim was attached by nails, he was laid on the ground, with his shoulders on the crossbeam. His arms were held out and nailed to the two ends of the crossbeam, which was then raised and fixed on top of the vertical beam. The victim’s feet were then nailed down against this vertical stake.
So someone carrying their own cross was the standard practice, and it would have been a relatively common sight in Jerusalem.
Tzaferiz, V. (1985) Crucifixion — the archaeological evidence. Biblical Archaeological Review, 11(1), 44-53.