How does supply & demand effect bond yields vs equity prices?

Will preface this by saying I'm not a finance guy but trying to learn more, although I majored in Finance back in the day...

Bonds confuse me because i feel like yields can change for many reasons and some of them seem contradicting. As I understand it, bond yields will fall when the stock market is down because investors take their money out of stocks and buy bonds, thus raising the price of the bonds from the demand. So I see headlines of bond yields rising, which is causing investors to flock to the bond market which decreases equity prices as people sell stocks to buy bonds. But wouldn't this immediately cause bond prices to rise again due to demand? i feel like the stock market and the bond market should have inverse reactions. Or that theoretically they would be in equilibrium due to arbitrage? Idk my brain is in a pretzel from reading to much investopedia when i should have been paying attention to Econ classes in college. Why do bond yields keep rising as investors buy more bonds? I'm sure I'm either misunderstanding or missing something obvious, but I can't make the connection. Can someone please explain?