Here is how I interpret these sentences:
I suggested that she bought a car
This would be equivalent to "I gave the impression that she bought a car [even if that wasn't the truth]." In this case the speaker was not addressing the presumed buyer of the car ("she") while doing the suggesting.
I suggested that he gives up golf.
I think this is a less formal/"correct" way of saying "I suggested that he give up golf" - meaning "I told him [or someone else] that he should give up golf." (Normally the present subjunctive uses a bare infinitive form of the verb.) The speaker may or may not have addressed the other person ("he") while doing the suggesting.
EDIT: For sentence 2, I am now not sure that thinking of the suggestion as possibly being directed towards the other person ("he") actually makes sense. The meaning of "to suggest" that makes sense to me in both sentences is "to imply as a possibility" (to anybody, not necessarily the "he" or "she" spoken about) rather than "to advise" (where the "he" or "she" is the recipient of the advice). This way, the subjunctive mood isn't required, and the sentences do not sound ungrammatical. The second sentence therefore would have a meaning more like "I implied that he gives up golf [on occasion; whenever someone tells him to; etc.]"
EDIT 2: I have learned from the OP that the rules concerning use of the subjunctive are different in British English from (my native) American English. The original sentences are in fact valid expressions of the act of giving advice. If you're talking to an American audience, you might be met with confusion, but if the audience is British then I suggest you leave the sentences be. (see what I did there? :-p)