Is it normal to not "own" your bitcoins? Are 6% transaction fees normal?
There are both custodial and non-custodial online services as well as apps. If it's important to you to control your own funds, you should research the account model your service provider uses and avoid services that manage your funds for you.

Bitcoin transactions are not paid respective to the amount that is being transferred, but rather in regards to the blockchain space that you purchase in order to have your transaction confirmed. Transaction fees have been much higher in the past months than previously, as the supply is capped and demand has been increasing. Personally, I'm hoping that a protocol upgrade by the name of Segregated Witness will be activated soon, as this will increase the blockspace as well as provide access to a payment layer that I expect to work as a valve to curb demand.

For you personally this means that fees are more or less the same size for whatever amount of money you transfer. If you make large payments, the fees will be minuscule, if you make small payments, the fees will be a significant port of the transferred value. Naturally, this is making Bitcoin unattractive for smaller payments.