Not heard of the HDMI and DVI being linked on the R9 270X, and it's not a standard feature of the card (
https://www.techpowerup.com/reviews/MSI/R9_270X_Gaming/3.html mentions all 4 outputs being usable at the same time). Which make/model of R9 270X do you have as I'd hate to lead you down the wrong path if yours is different?
What make and model are your monitors, and do you ever hot swap them (unplug/replug when the PC is power on) or are then always connected at startup and remain that way?
The reason I'm asking is that the R9 series has only two clock generators (as have all AMD cards since the old HD series at least), and any DVI/HDMI/VGA output needs a clock generator, so you're usually limited to two DVI/HDMI/VGA outputs on the card. However, the R9 series has a special feature where if two monitors are the same make and model, and are plugged in at startup, they can share one clock generator - hence my three E2409HDS monitors are connected by DVI, HDMI and DVI and all three work fine. DisplayPort outputs do not need a clock generator so can be used if both clock generators are already in use. Passive DP adapters to DVI/HDMI/VGA still use the GPU's clock generator, but Active adapters include their own so don't use one of the GPU's.
So depending on what monitors you have, and what connections they have, we might be able to get all four working from your R9 270X without needing the extra GPU.