If I short a lot of shares, will it make the share price drop even further?

... so suppose I short a lot of shares thus increasing the supply

For every buyer there is a seller and vice versa. So shorting does not increase the supply, aka the float. It proportionally increases the number of shares in long and short positions. Read my answer here to see why.

If I short a lot of shares, will it make the share price drop even further?

Maybe, maybe not. It depends on what the other participants in the market's auction are doing. Read my answer here.

You are sort of correct, but you have confusion about "shorting".

1. In your question, simply forget about "shorting"

2. Now we ask, "If I have a lot of shares and sell them, will it make the price go down?"

3. The answer is very simple: Yes.

The next thing you ask is:

1. What determines the price of shares?

2. This is very simply answered: bids and offers determine the price of shares. Nothing else. If a lot of people want share X, the price goes up. If a lot of people want to get rid of share X, the price goes down. It is literally pure and simple. There is nothing else, whatsoever, involved.

I think the real crux of your question is:

do people have a general consensus on what the price of a share of a company should be?

The short answer is: That's called the current share price.

Long answer: If you are shorting a stock, you're contributing your voice (in the form of your wallet) to the consensus, saying you think the current consensus is wrong, and it should be worth less. If a lot of people end up agreeing with you, the share price moves down; if they don't, it stays steady or moves up. For small shorts, your voice is likely lost in the noise, but if a major investor or fund makes a big short bet on a company, that can absolutely drive down the price of the company on its own in that:

1. They have to sell to someone, and
2. At any given point in time, there are a limited number of buy orders submitted or queued at a given price

If company X's shares are selling for $10, it just means at least one person was willing to pay that. If you try to sell 1M shares short, and there aren't enough folks willing to pay $10 for shares to make up that order (nor any trading algorithms willing to swoop in to match your ask with a matching bid), you have to work down the line of bids until:

1. You complete your order at whatever the one-millionth highest bid for a share is, or
2. You hit your own lower limit on what you're willing to sell for (which means you don't fill your order)

Whichever happens first, when you stop, that's the new share price (until someone else buys at a new price). But it doesn't mean you can turn around and sell the 1M (or maybe fewer) shares for that new price and make a quick buck; you have the same problem going the other way now, where you've got to bid enough to buy each share, not just enough to buy the first share.

Personal investors rarely see this effect (when you buy or sell in lots of 100-1000 shares, you usually get matched to someone selling that many shares and pay a single price for all the shares). But institutional investors absolutely do; if nothing else, high-frequency trading algorithms will try to identify your short selling pattern, figure out how much you're getting rid of (obvious if you submit the whole order in bulk, but oftentimes large investors break up the orders to make this less obvious; you don't want the people buying to know you're going to be immediately driving the price down further, nor to know the lower limit you're willing to sell at, since they'd just reduce their bids to match), how low you're willing to sell for, and try to get ahead of you by selling before you get there for a higher price, then buying (possibly to cover their own short if they had no shares of their own) from you for a lower price a moment later.

All the fundamentals of a company inform the consensus (among other things, companies that issue regular dividends can be valued, if on nothing else, by the effective interest earned on the money invested in them, and the confidence that said dividends will continue and/or increase), but the actual share price is always "whatever people are willing to pay" (as evidenced by what someone actually paid most recently). If someone sells at one-tenth the current price for whatever reason, technically, the share prices drops to one-tenth the current price. But in practice, that one-tenth sale never happens (why give up 90% of the value?), and if it did, it would almost always be followed by the lucky buyer immediately selling it on to someone else whose bid was at the old price, or just below it, and making an immediate profit of close to 900%, or someone else selling at the "normal" price in an unrelated transaction restoring the price.