If you aren't the sole controller of your private keys, you don't have any bitcoins.
Possession isn't necessary for ownership.
Just because you transfer custody of your coins to someone doesn't make the new holders of the coin the new owners, just like when you take your kids to school it doesn't make the teacher the parent.
You entrust your fiat / cryptocurrency to exchanges for the specific purpose they set forth in their agreement -- essentially to safeguard until you have removed it from their system.
Bitcoins aren't children.
Bitcoins are a cryptocurrency. This means that they rely on cryptographic proof of ownership. That cryptographic proof is provided with a digital signature from a private key that is cryptographically linked to the bitcoin address. When you "send" bitcoins, what you
literally do is digitally sign an agreement that transfers ownership of the value to the controllers of a specific set of private keys.
If you don't have exclusive control of the private keys, then you don't have the bitcoins.
Any time you send your bitcoins to an address where you don't have exclusive control of the private keys (such as MtGox, CoinBase, BitStamp, BTC-E, localbitcoins, etc), you are making a donation to whoever has control of those private keys.
The bitcoins immediately belong to them and not to you. They are no longer "your" bitcoins. In return, most of the people who run these websites will offer you a promise that they will send an equal amount of their bitcoins wherever you ask them to whenever you ask them to. You need to decide if you trust them to deliver on such a promise, keeping in mind the possibilities that they could have security flaws, could find their assets seized by government agencies, could be tempted to keep the bitcoins for themselves, and a variety of other situations that could cause them to fail to live up to the promise provided.
You are not "entrusting your fiat / cryptocurrency to exchanges", you are
purchasing a promise. You are trusting the exchange to deliver on that promise. There are many untrustworthy individuals and many untrustworthy organizations in the world. Be careful about purchasing promises from untrustworthy entities. The promises are often not worth what you pay for them.