How do I connect a MacBook Pro 2013 with Thunderbolt 2 to a new MacBook Pro M4 with Thunderbolt 5?

You could do it with an Apple Thunderbolt 3-to-2 adapter ($50) and a Thunderbolt 1/2 cable ($30+, harder to find). But instead of doing that, I'd suggest migrating via a backup – Time Machine, CCC, or SuperDuper!.

You may already have a backup on hand that you can bring up to date right before the migration. If not, you could take the $80+ that cabling would cost, and put it towards the purchase of an external drive that you could use first to do the transfer, and then to back up the new Mac.

Note that it is easy to convert between USB-A (USB) and USB-C (USB) in either direction. So an external drive that can connect via USB – whether USB-A (USB 3.0) or USB-C (USB 3.1 Gen 2) – would be the easiest to connect first, to the old MacBook Pro; and then, to the new one.

The best way to migrate data from one Mac to another is by using a Time Machine backup on an external drive.

Not only does it do away with the requirement of unusual and/or expensive cable connections, it also gives you the opportunity to continue to backup your new Mac to an external drive.

If your stuff is importantly to you, then you have to back it up.

Migrating your data is always Very slow.

There is more data than you think, and

there is more computing around doing this than seems reasonable and

the effective transfer rate is always slower than expected.

But you endure it ONCE.