is it normal for ATC to ask an airplane 10NM from touchdown to make any major turn (e.g. a turn of, say 30 degrees or more)?
Very regular, for all sizes of aircraft up to and including 747's.
It will be most often that the closing course to a localiser will be "offset" by up to 30 degrees, and the aircraft will be expected to establish on final approach course when intercepting. That can occur as late as 4 miles, though that is expecting more of the crew. At 10 miles, you could easlily get up to 90 degrees of change, depending on the approach routing. If you were flying a procedure approach, then there could be a complete reversal of almost 180 degrees at somewhere around 8 to 12 miles, occasionally closer to the airport. That's not a problem to do in commercial aircraft.
A "standard" hold before approach due to congestions is a 4 minute "racetrack" around a fixed beacon point. 1 minute on heading, reversal of 180 degrees over a minute, 1 minute on reciprocal, 1 minute change back again through 180.
If you're having problems with this, what speed are you trying to do it at?
Most approach speed work will be done at less than 250 Kts, and close in, more like 200-210 Kts, depending on type, smaller aircraft, it can be considerably slower.
Hope that helps