For the chassis refresh, I wouldn't worry about it unless the handling is just so bad, you cannot tune around it.
Now for the turbo...long story short, turbo lag from the larger turbos can upset handling. Keep an NA car NA if you can, use a lag-free supercharger when possible, or only use the largest turbo upgrades for speed runs -- with the smaller ones for the track.
A turbo only slightly affects handling, and it mostly depends on how large the turbo is, and how powerful the base engine is before the upgrade. It was very noticeable in GT3 and 4, while it's only slightly less noticeable in GT5 (but still there).
A low power engine with a large turbo will suffer turbo-lag coming out of corners, spinning the drive wheels at maximum boost pressure kicks in, and either understeer (FF, some 4WD, and some RR) into a wall, or oversteer (MR, FR, some RR, and some 4WD) into a wall. More powerful engines can still suffer from the lag, but it's not as bad.
I saw a lot of this in GT4, with EGs and AE86s equipped with Stage 4 turbos and no assists. You get great corner entry speeds, as well as speeds on long straights, but corner exits became a handful.