Oil is one of the most fundamental economic inputs when it comes to determining costs and thus inflation.  Inflation is one of the most fundamental inputs when it comes to determining the price of bonds.  It certainly stands to reason that big moves in oil could translate to at least some movement in the bond market.  While that is indeed true in many cases, there is often a lag time for the correlation to play out.  Beyond that, both oil and bonds frequently have other things to worry about. Sometimes those things have similar effects on the two markets, but other times, only one of the two is majorly affected.

The red highlighted areas show notable instances of oil and bonds moving on completely different agendas.  When we bring in indicators like the value of the dollar and China/US stock indices, we have a clearer picture (to whatever extent anything can be clear when there are this many lines on the chart!).  The important thing to notice is that the 2014 oil price crash was a function of the dollar.  Ongoing dollar weakness kept oil prices generally low through mid-2017 when dollar strength combined with economic expansion to push oil prices higher.  It wasn't until the late-2018 "global growth concerns" resurfaced that oil began to slide again.

Most recently, oil has been on the comeback path.  This isn't a mere factor of some of the Iran/supply/sanction news, even though financial media unfortunately tends to offer such overly-reductive reasoning.  The fact is that global growth concerns have been ebbing for most of 2019, as evidenced by the rebound in equities.  Bonds had avoided the typical ill effects of such a rebound due to the friendliness of the Fed and the fact that bonds were arguably a bit weaker than they should have been by the end of 2018.

At present the recent weakness in bonds is best explained by that same easing of global growth concerns.  Oil is incidentally benefiting from the same thing.  Could the rise in oil prices be compounding the rise in bond yields?  Yes, but it's fairly far down the list of market movers we're watching.