Nobody ever claims that. Certainly not anyone who is serious about Peak Oil.
Just the fact of supply falling short of demand is dire enough. Production has stopped expanding, while demand has not. Most of the major oil fields have peaked and are in decline. Saudi Arabia is playing like it could increase production rates "if it wanted to," but this should soon prove to be a coy pose.
The Carioca field in Brazil could be fairly big, perhaps 5 billion barrels, but it's early to tell. Tupi, also recent, is of comparable size, but both are very hard to get at and will take a long time to develop.
Estimates for Tupi put production at about 100,000 barrels per day at first, but it will take until 2020 or so to get up to 1 million bpd.
These figures just barely qualify the fields as "large." Current oil production worldwide is around 86 million bpd. It's been flat for three years, and will not be going up. Decline is inevitable, starting right about now, and these new discoveries will, at best, simply prevent the decline from being quite so steep.
Yes, we need solar and wind. "Clean coal" is an oxymoron right up there with "healthy cigarettes," but we'll probably see a lot more coal anyway. Nuclear? Well, "a little" is definitely the operative term. But we have to keep in mind that none of the alternatives, or any combination of them, will be able to make up for diminishing oil reserves.