There are some baby quantum computers with very small processing power, that they have constructed in labs, but nothing that is better than a classical supercomputer.

Yes the internet will be affected, and many online services but Bitcoin will not be if you follow the advices in this artcile and the next one:
https://steemit.com/bitcoin/@profitgenerator/bitcoin-security-overview-and-quantum-computers

• Bitcoin Mining is Quantum Proof
• Unspent Addresses are Quantum Proof
• Spent Addresses are not Quantum proof, but ECDSA can be soft-forked and replaced with something better later on after the Segregated Witness package goes online in Bitcoin.