What motherboard is this for? The Asus Z790a in your sig has the required internal USB3/C headers, so I'm guessing you aren't talking about that one.
If you have spare PCIe slots/lanes, it is possible to add the proper internal headers (and some extra ports via PCIe add-in cards) through PCIe and not lose performance.
OCUK have this on the site, which luckily uses a single PCIe2 x1 lane (so easy to fit in to older motherboards) and provides you with 2 rear USB 3 ports and most importantly, the internal USB 3 header for your case to connect to. If using on a PCIe2 x1 slot, you'll only be able to get the max speed of 5gpbs on a single USB3 device at any given time, but it's better than bottlenecking it to USB2 speeds (alternatively if plugged into a PCIe3 or above, then you'll have the bandwidth for more simultaneous connections). If you're lucky enough to have a PCIe4 x1 slot or even just additional PCIe 4 slots (which is rare, even on the latest motherboards), then you can get the full performance.
Order AXAGON PCEU-232VLS PCIe Adapter 2+2 x USB3.0 UASP VIA + LP, 15-pin SATA Controller Card now online and benefit from fast delivery.
www.overclockers.co.uk
Startech also make the same part, but way more pricey, several times the cost of this one. I can't link it here since it's on a competitor website, just google the following part number: PEXUSB3S2EI
For the USB-C, Akasa does an adaptor that takes 4 lanes of PCIe3 or better to give you the full bandwidth connection on an internal header.
Akasa branded products including cables, cases, card readers, coolers, external enclosures, fans in many colours, hubs, and power supplies |USB 3.2 Gen 2 Internal 20-pin Connector to PCI Express Expansion Card | |
www.akasa.co.uk
Otherwise, if you're willing to lose performance, there are internal USB3 to internal USB2 adaptor cables like this one:
Order Akasa Internal USB 3.0 Male to USB 2.0 Female Adapter Cable now online and benefit from fast delivery.
www.overclockers.co.uk
Unfortunately there is no internal USB-C to internal USB2 adaptor, but the closest is an internal USB-C to internal USB3 adaptor.
Order Kolink USB3 to USB Type C Adapter now online and benefit from fast delivery.
www.overclockers.co.uk
There is also similar for USB2, if you want more USB2 headers or want to make use of those USB2 internal connections.
My recommendation, don't overlook the unused PCIe slots, they can be very helpful in cases like these for the best solutions (go with the PCIe add-in cards). Else go with the header adaptors, which will probably do the job, but I personally can't vouch for them.