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While winding a core of the transformer some turns cross other turns. Will this affect the magnetization produced by the core?

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For a toroid (a closed magnetic core)

Providing that the windings are electrically insulated there is no significant effect of them crossing each other. More turns means more inductance whether they are wound side-by-side or on top of each other. You could argue that an overlapping wire is slightly longer because of its radius and this has slightly more resistance of course. You can also argue that overlapping wires produce more self-capacitance and this can lower the self-resonant frequency of the wound component. But, given your title states an "electromagnet" this is of no consequence because they are generally run at DC or low frequencies.

For open magnetic cores

You get a better concentrated magnetic field with overlapping wires than with side-by-side wires up to a certain point. This is because side-by-side turns at extreme ends of an open core do not readily couple with each other. At the other end of the extreme, if you could pile turns on top of each other, there comes a point when it's sensible to also wind them side-by-side (in order to maximize the magnetic field for the minimum number of turns used).

Anomaly in the question title

Is there any effect on efficiency of electromagnet if winding wire cross each other during winding?

A closed toroid is (and never can be) an electromagnet. An electromagnet is used to attract ferrous metal and requires a gap in the core that forms north and south poles so that iron (e.g.) can be physically attracted.

While winding a core of the transformer some turns cross other turns.

A transformer is not an electromagnet so, there appears to be significant confusion in your question. If it is a transformer then it could be used at high frequencies and lower self-resonant frequencies (due to overlapping windings) may be of significant interest in how the device performs.

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When wound round a high permeability core, crossing wires in a winding simply takes up more space than laying the wires uniformly.

Other things being equal, with the same space available for winding and the same number of turns, a scramble-wound winding has to be designed with thinner wire than a neat winding. This means it will be higher resistance, and so less efficient. It may also have poorer thermal conductivity through the winding to the surface, so may run hotter internally.

With a low or unity permeability core, there will be subtle effects on coupling coefficient and self capacitance, which in precise applications may need to be controlled by precise winding.

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