I was reading a article Link It stated that lower bypass ratio jet engines have higher thrust than high bypass ratio engines, but the boeing 707 has 4 of these low bypass ratio engines and only travels about 600mph. Most modern high bypass ratio engines travel at about 514mph(push boeing 737-800 aircraft at an average of this speed) and only have 2 of these high bypass ratio engines Boeing 737-800 engine.I am having trouble understanding how this works.
Boeing 707 was initially designed to use the P&W JT3C turbojet engine. Early versions of the '707 (like the one in the first picture of your question) and C-135 (including KC-135A) used those engines. Wikipedia reports a maximum thrust of 12,030 lbf.
The JT-3D is one of the firsts early low-bypass ratio turbofans (1958/59). It's based on the JT3c and, according to Wikipedia, has a bypass ratio of 1.42:1 and a maximum thrust of about 17,000 lbf.
The CFM-56 first run was in 1974, more than 15 years later. The CFM-56 is a high-bypass ratio turbofan, with a bypass ratio of 5.5:1 and a maximum thrust of 19,500 lbf for the CFM56-7B18 model that powers the Boeing 737-600 as reported on Wikipedia.
So the answer to your question is that at the time the Boeing 707 was designed, the only available engines were turbojets with a limited (for today standards) thrust.
The article you're asking about refers to modern jet engines. Today low-bypass ratio turbofans are used mainly in military supersonic aircrafts, were fuel efficiency is less important than thrust and supersonic efficiency. The opposite applies on modern civil aircrafts.
Engine technology has improved tremendously since the days of the Boeing 707. Today's engines are larger, more powerful, more efficient and quieter than their predecessors.
The engines used in first production Boeing 707s (from 1958), the Pratt & Whitney JT3C turbojet, developed a thrust of 50KN (11200 lbf). This engine was the first one (in US) to develop more that 10000 lbf. The same engine (Pratt & Whitney J57) was used in the B-52B stratofortress.
This engine was selected as it was the best available (in fact, the J57 designers won the Collier trophy for their efforts). In fact, the engine was so successful that aircrafts were designed around it- half of the century series USAF fighters used them, as did 707's competitor, the Douglas DC-8.
As for Boeing 737, which had its first flight around 10 years later, the Pratt & Whitney JT8D used was a low bypass turbofan engine, having a bypass ratio of 0.96:1. Again, it was developed from a military engine, the Pratt & Whitney J52 (which powers Northrop Grumman EA-6B Prowler, among others).
One pattern we can see clearly is that most of the early civil aircraft engines were developed from military turbojets. The reason was that only these powerplants could generate the required thrust at that time and today's high bypass turbofans were not even in the picture.
However, as the airliners became fuel conscious and noise became a major issue, high bypass turbofans were developed exclusively for civil use and the development of civil and military engines diverged. With time, the bypass ratios have increases with decreases in fuel consumption.
As a comparison, the Rolls Royce Trent-1000 engines used in Boeing 787 can generate 350KN (78000 lbf) of thrust, more than 6 times that of JT3C. So, for meeting the thrust requirement of the four engined Boeing 707, you don't even need a single 787 engine.
In the context, it is more useful if you compare the engines used in the same aircraft over the years. The most powerful engine used in the 707, the Pratt & Whitney JT3D-7 increased thrust over JT3C by over 50%. In case of 737, the 737 MAX uses CFM International LEAP, which doubles the thrust on the PW JT8D.
As for the article, what I get is this: For a given power available to the bypass fan, it can be used in either the two ways- take less air and increase the pressure of it more or take more air and have a lesser increase in pressure.
In general, the military turbofans have a lower bypass and higher thrust to weight ratio. For example, the Pratt & Whitney F119 used in F-22 Raptor has a bypass ratio of 0.3:1 and a thrust to weight ratio in excess of 9, while the General Electric GenEx used in Boeing 787 has a bypass ratio of 9.6:1 and a thrust to weight ratio of 5.