Are there any estimates as to how many people witnessed the crucifixion of 'Christ'?

We need to differentiate between 'witnessing something' and 'looking at ' it with the mindset of a passerby. Public execution has always been a subject of curiosity among the onlookers, even if the convicts are total strangers to them. Now, the big difference between crucifixion and the other methods like hanging of execution is that in the former, it can never be predicted when the convict is likely to die. Some convicts would hang for days on end, with the gall administered to prevent dehydration. It is quite doubtful if the scene of crucifixion would attract many who would stay put to watch the victim die. In the case of Jesus, he was given the colour of someone who opposed the Roman Empire. How many Jews would dare standing at the foot of the Cross to invite Roman wrath ? Jesus' own disciples except John had fled for their life ! The remaining few were Mother Mary, the woman disciples and a few gentlemen who had some social positions. Of course, the soldiers likely numbering four, with their Commander in Charge, were there on duty, not to mention of the crucified thieves . Altogether, there may not have been more than fifty witnesses . The passerbys who stopped for a moment or two to read the inscription , and then proceeded on their business, cannot be treated as Witnesses. Can they ?

Roman Practice We know from the habit of Emperor Nero that he delighted in lining the roadways with crucified or staked Christians and setting them on fire. And it was a practice of the Roman military establishment to line roadways with the convicted who were sentenced to be crucified to serve as an example to the population so they would think twice about committing crimes or acts of treason!

The Bible mentioned the crowd of priests and scribes, the women from Galilee, the Roman execution detail, Simon from Cyrene, the disciple John, and curious on-lookers, were all at this crucifixion. (Matthew 27:35-36, 41-43, 47, 55-56, 57-61, Mark 15:21, 31, 35, 39-41, Luke 23:27, 35-36, 47-49, John 19:25-27)

Pilgrim Train But beyond these people, because of the practice of placing the crosses along the roadway, AND because it was the Feast of Passover (and Unleavened Bread),many thousands of Jewish pilgrims would have seen Jesus's crucifixion! Flavius Josephus, may or may not have always been accurate with numbers, but he attests to millions attending the Passover (q.v. in loc.). Tens of thousands traveling the road into Jerusalem would have witnessed this crucifixion!

Recall the comments of the two disciples on the road to Emmaus, about all the commotion caused by this crucifixion. (Luke 24:18) And this "stranger" who joined them should have known about it all. "Are you just a new visitor here?" (If only they knew who they were talking to!!!)

Unique Scene Would pilgrims have just passed on by this scene without noticing anything of importance? No way! The sneering crowd, the weeping relatives and women, the mocking priests, the cursing convicts...would all have attracted attention. This was a noisy crowd. Jesus called out with a loud voice, 'Father, into your hands I commit my spirit!' (Luke 23:46)

Those who passed by hurled insults at Him, shaking their heads, and saying, 'You who are going to destroy the Temple and build it in three days, save yourself. Come down from the cross, if you are the Son of God!'

They would also have been drawn close to try to read the sign above the body of Jesus that Pilate put there. (Matthew 27:37, John 19:19) AND if all this commotion wasn't enough, the many pilgrims would have had their attention gotten by the sudden darkness at high noon, and a massive earthquake that split open rocks...and graves...and the Temple veil!!! (Matthew 27:51-52, Mark 15:33) Walking around in darkness, with the ground under your feet shaking, would have caused any people to wonder, and ask about what they had just witnessed, and who was being crucified!

Estimates We have no exact number of those who "witnessed " the Crucifixion. But as far as "estimates" go, it is safe to reckon that there were tens of thousands. There were hundreds who knew what was going on, and thousands who were about to find out in a shocking way. Breaking News: the Prince of Life is Dead! Stay tuned for further developments.

There are two answers so far, both are good, and each has a very different number, both of which could be right.

One first needs to know how common crucifixion was in that time and place.

• If it was common, then most people would simply ignore it, and only those that actually knew the victims would be there.
• But if it was rare, then yes, its possible that 10s of thousands looked for at least a few seconds.

But even if the second case applies, there's still the problem of knowing exactly what "witnessed" means.

• If someone was walking by and momentarily glanced up, does that count as witnessed?
• Or does witnessing require stopping, observing, being emotionally affected by what was happening?

The other problem is that we don't know for sure where Golgotha was, and so don't know how many people could observe at the same time.

• If only a dozen people could get a good view, and 5 minutes were required to be a witness, then the maximum number of observers possible during those six hours would be fewer than 1000. And even less if we consider that toward the end most people would be hurrying to get to their destinations before the sabbath began.
• If it was on a hilltop near a city gate, then hundreds of people could observe at the same time, and if looking for only a few seconds counts as witnessing, then hundreds of thousands of people could have been witnesses.

The OP can clarify what is meant by "witnessed". But without knowing how rare these public executions were, and how visible the location was, it's impossible to give a reasonable estimate.