Empathy might not work that way. See: Notes on Empathy.
For one thing, we seem to be wired to empathize more with people in the in-group than people in the out-group. For another, once we begin to see a conflict through the lens of empathy, we tend to adjust our interpretation of the evidence so as to share the interests and bias of whomever we first began to empathize with in the conflict. In short: empathy ought to be approached with caution.
Today is the fast of Av, when Jews mourn the destruction of their temples in Jerusalem and their exile. We fast, sit on low stools, and read the book of lamentations (Eichah).
This book, traditionally ascribed to the prophet Jeremiah, consists of 5 poems mourning the destruction of Jerusalem and the horrors perpetrated by the invaders.
They were most probably written by 5 different, anonymous, authors, all of whom lived during, or shortly after the conflict.
The poems are visceral and brutal:
Will women devour their own offspring, children that are petted? Will priest and prophet be slain in the Sanctuary of the Lord?
In the streets, on the ground lie young and old, my maidens and my young men have fallen by the sword; You have slain them in the day of Your anger; You have slaughtered without mercy.
When read in the traditional dirge-like tune, whilst sitting on the floor, hungry and thirsty, it is hard not to connect with the victims of that tragedy so long ago. You can almost see bodies lying in the street, buildings destroyed, people starving, an enemy cruel and hard who shows no pity to woman or child.
But these emotions are as pointless as they are powerful. Both victim and perpetrator lived 2500 years ago. There is no-one left to save, and no-one left to bring to justice. No-one to avenge and nothing to wreck vengeance on. Our empathy is guilt free, it demands nothing from us.
Unfortunately there have been no end of conflicts since then, and every one of them has had atrocities as bad as those described in lamentations. Wikipedia lists no less than 10 genocides in the 21st century, many still ongoing. But we spend a lot less than 1 day a year thinking about them.
Today is also the 3rd day of the umpteenth conflict between Israel and Gaza. You almost certainly sympathise with the civilians who are currently living in fear, not knowing when the next strike will come, their lives subject to the whims of politicians who decide when the current round of tit for tat will end.
However depending on your political views those civilians are probably either Israeli or Palestinian. Rarely both. Why not? (And yes, the Gazans do have it much worse - they don't have bomb shelters, air raid sirens, or anywhere to flee to. But that doesn't diminish the suffering of Israeli civilians).
I live in Israel, just 20km from Gaza. Most of the people here are good people, much like everywhere else. They care about friends and even strangers. They give to charity and hold open the door for people in wheelchairs.
A Russian in Kiryat Shmone considers it a tragedy when they read about the death of a 5 year old Moroccan boy in Sderot 200 km away. But the death of a 5 year old Palestinian girl just a few km further in Gaza city is just the unavoidable cost of war. There might be some Israelis who consider it a good thing, but they are few and far between. For most, they just don't particularly care.
There might be some Palestinians who are happy when they kill a 5 year old Israeli, but for most it is again the unavoidable cost of war, possibly justified for its knock on affect in frightening Israelis and putting pressure on them.
For both sides Israeli/Palestinian lives simply don't count as much as Palestinian/Israeli lives.
And when both sides have finally finished flexing their biceps and settled down to lick their wounds till the next round of conflicts, nothing will change. Israel will continue blockading Gaza, and Gazans will continue living with an awful quality of life
Not because Israelis hate Gazans, nor because any solution is impossible. Israel could let Gazans live and work in Israel. But any solution would be expensive, and take a lot of time, effort and goodwill.
And there would be deaths. There will undoubtedly be more terrorist attacks if Gazan Palestinians had free reign in Israel, whatever security measures were put in place. But 100 Palestinian civilians were killed just a year ago in the last round of conflicts, and 1500 seven years before that. It is unlikely anywhere near that number would be killed in terrorist attacks, so on net we would have saved lives. But those are Palestinian lives we would have saved, and they simply don't count as much.
Perhaps the lesson of lamentations is that if we can empathize with people who lived 2500 years ago, we can empathize with those who live just a few kilometres away today.