your　are parsing it wrong.

車を運転する時はいくら注意してもしすぎることはない。

In parts,

車を運転する時は means "when you drive a car" and explains in what circumstances the latter part of the sentence applies.

いくら注意してもしすぎることはない

いくら = how much to / to what degree

注意 has two meanings. Either to warn someone or to be mindful of something.

して = て-form of する

も - this makes it a pattern いくらVても - "no matter how much". In this case, roughly "how careful you are"

しすぎる = し is the stem of する and すぎる is to do something too much. ~ことはない = there is no such thing as ~.

So for the whole:

When driving a car, there is no such thing as being too careful.

or

When you're driving a car, you can never be too careful.

You are parsing it wrong.

車を運転する時はいくら注意してもしすぎることはない。

In parts,

車を運転する時は means "when you drive a car" and explains in what circumstances the latter part of the sentence applies.

いくら注意してもしすぎることはない

いくら = how much to / to what degree

注意 has two meanings. Either to warn someone or to be mindful of something.

して = て-form of する

も - this makes it a pattern いくらVても - "no matter how much". In this case, roughly "how careful you are"

しすぎる = し is the stem of する and すぎる is to do something too much. ~ことはない = there is no such thing as ~.

So for the whole:

When driving a car, there is no such thing as being too careful.

or

When you're driving a car, you can never be too careful.