Late 2013 MacBook Pro Retina external monitor

That model appear to offer up to 2K resolution at reasonable refresh rates:

2nd Display Support: Dual/Mirroring* 2nd Max. Resolution: 2560x1600 (x2*)

Details:* This model supports a simultaneous maximum resolution up to 2560x1600 on two external displays via Thunderbolt.

Alternately, it can support a single display up to 2560x1600 via Thunderbolt and

a single 1080p display at up to 60 Hz, 3840x2160 at 30 Hz, or 4096x2160 at 24 Hz via HDMI.

data from:

EveryMac.com

By far the easiest way to cause poor performance, instability, overheating and crashing is to install ANY third-party speeder-uppers, Cleaners, Optimizers, or Virus scanners. or a VPN that you installed yourself.

The idea that a third party, with no special knowledge of the inner workings of MacOS, can somehow find a simple way to protect your computer — that is not already being done by MacOS itself — suggests that the MacOS developers are somehow "holding out on you". That is absurd.

You should remove any and all (other than Apple built-in) virus scanners, speeder uppers, optimizers, cleaners, App deleters or VPN packages you installed yourself, or anything of that ilk.

Third-party file Sync-ers such as DropBox, BackBlaze, OneDrive, or GoogleDrive can ruin performance, but are not inherently dangerous.

In general, supporting a large external display has only an imperceptible influence on Mac responsiveness.

That is because display data rates are so high, the servicing such a display using processing power is completely impossible.

Instead, a display-generator automaton is used for fetch the data from Display memory. The processor intervenes only once a screenful. To further reduce its impact your model MacBook Pro has 2 G Bytes of private display memory, so the impact of adding an external display is one lower.