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1. They started their show immediately.
2. They immediately started their show.
This one has a different meaning:
3. They immediately started to their show.
In older English, it would have meant "They immediately set off for their show". However, that meaning of "start to" is no longer current; and it wouldn't make much sense, in this context!
Best wishes,
MrP
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start to their show/ start their show 
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