I can see here two or three possible reasons:
1. problem with the phone: it would mean OsmAnd gets the wrong position (it "sees" itself further back in the road than it is, the GPS info is delayed)
2. problem with the phone: the position correct but OsmAnd for some reason react too late (RAM?)
3. problem with the route itself.
I see as probable either 1 or 2, what phone are you using? You might be able to see it if you zone at your current position - it would show you at the place the phone sees you.
For 3:
How are you navigating? Along the downloaded route, or do you leave OsmAnd navigate you?
If you load the route and inspect it, does it correspond with the roads you see on the screen?
Majka