The first key to the answer is to note the Jews were incorrect in asserting Jesus was claiming equality at John 5:18. Jesus disabuses them of this notion immediately after.
"So Jesus replied, “Truly, truly, I tell you, the Son can do nothing
by Himself, unless He sees the Father doing it." (John 5:19)
Jesus is not equal to the Father, because the Son can do nothing by himself. That's the first error the Jews here were making, and the basic one.
Yet, the error is compounded by a second error, and so the claim about Jesus at 5:18 is a composite error. The second key, then, is to note they were making an error in thinking that Jesus wasn't just inappropriately claiming equality (perhaps informed by, for ex., doing what he wanted to on the Sabbath), but was committing blasphemy by simply claiming to be 'the Son of God' ("but He was even calling God His own Father, making Himself equal with God").
'The Son of God' was a way of referring to the Messiah. Consider not just John 1:49, where Nathanael uses the term and pairs it with the 'King of Israel', and Psalm 2:17 where the Son of God is described similarly, but the high priest at Mark 14:61.
"But Jesus remained silent and made no reply. Again the high priest
questioned Him, “Are You the Christ [i.e., the Messiah], the Son of the Blessed One?”
To claim to be 'the Son of God' was to claim to be the Messiah. This would be blasphemy, except in one case. They believed Jesus claiming to be 'the Son of God' was blasphemy because Jesus could not be the Messiah. Again, they were simply wrong - they misunderstood the prophecies about the Messiah and thought they would recognize the Messiah when he came. The Messiah couldn't be a son of a carpenter from Galilee, much less someone who would be critical of themselves! Rather, he was a great King, who would conquer their enemies - not judge the Jewish elite themselves!
Indeed, this is what does it for the high priest at Mark 14:62, after Jesus has been asked if he is the Messiah.
"“I am,” said Jesus, “and you will see the Son of Man sitting at the
right hand of Power and coming with the clouds of heaven.” 63 At this,
the high priest tore his clothes and declared, “Why do we need any
more witnesses? 64 You have heard the blasphemy. What is your
verdict?” And they all condemned Him as deserving of death."
The claim to be sitting at the right hand of Power is judgment language, drawing on Psalm 110 (the clouds language draws on Daniel 7, a Messianic prophecy).
"“Sit at My right hand until I make Your enemies a footstool for Your
Who was Jesus' enemy at that moment? The high priest. So Jesus here is saying to the high priest that not only is He the Messiah, the eternal King and Son of God, but that He would soon be sitting in judgment of the high priest and the others judging Jesus. This is highly blasphemous (unless, of course, it's true). (Jesus here is not claiming to be God, nor does the high priest's charge of blasphemy require that. To claim to sit at the right hand of God is not to claim to be God. The basic logic of the sentence makes that clear. If you are at the right hand of someone, you are not that someone. Compare Mark 15:10 - Jesus was handed over to the Roman authorities by the Jewish authorities because of envy. They were envious of a man gaining popular support as the Messiah - not as God.)
Putting this all together, the Jews Jesus was talking with throughout John made various mistakes. They thought Jesus was claiming to be equal to God (he wasn't - he could do nothing by himself), thought he couldn't be the Messiah (he was), thought he was 'declaring himself to be God' (John 10:33) when he was declaring himself to be an icon of God (or God's representative who had been sent by God, John 10:34-36), and thought they themselves couldn't be the ones who would be judged (they were).