I was able to test this myself and confirm that a few Reddit threads, including this one, match my observations.
If you buy a tile, it does not reset the culture progress towards acquiring a new tile. However, it does increase the amount of culture required to claim the next tile, but all progress is retained.
Comment from the thread:
It doesn't reset growth.
However, if you're at, say, 50/60 culture needed to expand, and you'll expand, in, let's say 2 turns, and you buy a tile... any tile... suddenly you're at 50/120 culture to expand, and you'll not expand for 14 turns.