Here is something from a New York Times article January 13, 1920

That professor Goddard, with his ‘chair’ in Clark College and the countenancing of the Smithsonian Institution [from which Goddard held a grant to research rocket flight], does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react — to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools.” (secondary citation)

That article started a trend which caused no end of long term trouble for Goddard's research. It wasn't the bad science of the article which caused the most damage. It was that a major reputable news source so ridiculed the research. Other newspapers joined in citing complaints of safety, noise, or that a person's body could not function in space. Rocket research became something of fiction to be laughed at in real life.

That turned around during World War Two when German rockets got a lot of attention. German scientists were brought to America where they tested captured V2 rockets. They became a political hot potato when one of those rockets went off course and hit a cemetery in Mexico. The Soviet Union seeing that American bombers were so far ahead of them saw American weakness in missiles and took the next step big time.

Your question is based on a misconception, which is that there was any sort of "space race" before the Soviets launched Sputnik. The US had some fairly low budget research programs, such as Project Vanguard: https://en.wikipedia.org/wiki/Project_Vanguard Unlike the Soviet programs, they were regarded as purely scientific*, and so did not use ICBMs as launch vehicles.

*There were also separate secret military programs to develop spy satellites, under the aegis of the military.

I think the premise of the question - that the US ought to have a technological superiority over the USSR, as it did in other areas - is quite sensible here, so I will address the Soviet side of the story.

• for some context, Soviet Russia happened to have it own vast talent in STEM (as witnessed e. g. by Nobel prizes to Tamm-Frank-Cherenkov and Landau), and an ability to concentrate this talent on things that were deemed important, sometimes taking extreme forms. The reasons for that is in part the legacy of old Russia, which was preserved and fostered in the early Soviet years. For example, Landau, then a young theoretical physicists, got funded research visits to Copenhagen and Cambridge when traveling abroad was our of bounds for most citizens. Later, STEM careers attracted a lot of talent, since business/finance/law alternatives promised much less than in the West, and also STEM elites were much less affected by Stalin's purges compared to other elites (although Landau and future space program head Korolev served prison time).
• in rocket propulsion, Russia had its own early pioneers, including theoreticians like Tsiolkovsky and Zander. Also, Korolev himself started experiments with rockets in 1932, approximately at the same time as von Braun, and worked on it until his arrest in 1938, at some point heading a research institute dedicated to this topic.
• one should not overestimate the value of von Braun and the V-2. First, it was a rocket based on the 1944 technology, and 1957 was ages apart relative to the speed of technological development at the time; new breakthroughs often require new people. Soviet Union also got hand on samples of V-2 and captured some scientists, and successfully replicated V-2 by 1950. One could compare the history of the atomic project; the first Soviet nuclear bomb was made in 1949 by copying Fat Boy's design obtained by intelligence, but by that time, the Soviets were already working on their own know-how, and in 1955, they actually were the first to test a combat-ready megaton hydrogen bomb. A similar story happened with strategic bombers, with B-29 replicated by 1947, but a much superior Tu-95 has seen the first flight in 1952 and entered mass production in 1956, it was so good that it's still in service.
• the amount of state effort concentrated on both the atomic and the missile projects in the Soviet Union was ridiculous. As one measure, while the Manhattan project was overseen by major general Leslie Groves, in the Soviet atomic project a similar role is attributed to Beria, the most powerful person in the USSR after Stalin. The head of Soviet army artillery marechal Nedelin personally oversaw many nuclear and missile tests and actually died in a missile test failure. BTW, contrary to the belief that R-7 of Korolev and its derivatives was the only Russian ICBR/space launcher project, that one was R-16 of Mihail Yangel, a simultameous, parallel and competing project. So, the budget and resources were really unlimited.
• just to reiterate, the ability of the atomic and missile projects to attract talent was enormous. Those in leading roles would be made academicians, the main accolade for a scientist in USSR involving lifetime salary and a lot of privileges, would get Lenin prize, the highest decoration of the state - the star of Hero of Socialist Labor and all in all would be about as rich and privileged as one could be in the Soviet Union, oftentimes a part of Nomeklatura.
• the reason the rocket project was deemed so important was, of course, already indicated - before Tu-95 that was the only viable way of striking the continental US with nuclear weapons, and thus achieving nuclear parity. It was a requirement to be able to carry the above-mentioned megaton charges, hence the effort was concentrated on quite heavy rockets from the start.

It was more of a back-and-forth.

You can build a narrative of one side out-pacing the other if you cherry-pick firsts, but their capabilities were very close. The timeline of first achievements is interleaved. Firsts grab headlines and demonstrate national priorities, but they don't show capability well.

The other side would often accomplish something similar only months later showing only very narrow capability gaps. For example, R-7 was the first successful ICBM and basis for the Soviet space program, the US had success with Atlas A less than a year later. Vostok-1, first human space flight and (just barely) first orbit, was followed a month later by Freedom 7 (first pilot controlled flight) and a year later by Friendship 7. Luna 9, first soft landing on the Moon, was followed a few months later by Surveyor 1.

They also both had very high failure rates. The Luna program had five failures (not made public at the time) before its first unmitigated success (Luna-2), meanwhile the US was losing Pioneer probes. The Luna program had a 35% success rate compared to 60% for equivalent US programs (the exact numbers depend on exactly how you count success).

Several of these firsts had close-calls which could have turned them into another failure. Vostok-1 had a partial separation failure on reentry. Gus Grisson almost drowned. Voskhod 2, the first spacewalk, had a suit malfunction.

However some areas did show a clear edge, for example imaging was dominated by the US because they wanted spy satellites. And one thing is clear, the US consistently made more orbital and deep space launches with a greater diversity of launch vehicles.

The civilian space race started when both the US and Soviets in 1955 announced they would launch satellites to celebrate the International Geophysical Year in 1958.

Prior to the start of the civilian space race in 1955, both nations were working on ICBMs. The Soviets built their successful R-7. They would focus their efforts on the R-7 eventually creating Molniya, and the wildly successful Vostok and Soyuz rocket families. This allowed the Soviets to focus their efforts, but also put most of their eggs into one basket.

In the US, rocket development became a turf war between the Army and the new Air Force (formerly the Army Air Force) with the Army focusing on tactical rockets, the Air Force on strategic rockets, and both sniping at each other to get funding.

The US was concerned with the optics of using a military rocket to launch civilian satellites, and that adding a civilian program to a top-priority military rocket might slow down its development. Military involvement would heighten then unresolved legal issues about overlying another country with a satellite.

The result was several families of launch vehicles being developed simultaneously by several different organizations. Slower to get started, but more robust. The Army's Redstone and variants Jupiter and Juno. The Air Force's Atlas. The Navy Research Laboratory's Viking sounding rocket was turned into the orbital launch vehicle Vanguard. Plus Thor which became Delta.

While the Soviets captured early firsts, the US was not particularly behind. Sputnik 1 was launched in October 1957, an 84 kg sphere that beeped, the simplest satellite. Sputnik 2 launched in November 1957 and was significantly more capable featuring telemetry, a Geiger counter, spectrophotometers, a dog life support (which unfortunately failed), and a TV camera. Meanwhile in the US Vanguard 1A failed in December 1957.

1958 would see the US rapidly catching up. 1958 saw 5 orbital launch attempts by the Soviets all with the R-7, and only one success: Sputnik 3 carrying numerous instruments.

The US had 23 launches across 6 rocket types with 5 successes and 2 partial failures.

• Explorer 1 put the first US satellite in orbit to catch up with the Soviets.
• Vanguard 1 was the first to use solar power.
• Explorer 3, same as Explorer 1.
• Explorer 4 studied the radiation belts.
• Pioneer 1 was an attempt to orbit the Moon, but failed to achieve lunar orbit. It had multiple instruments and was used to test communication relays.
• Pioneer 3 was intended to reach the Moon and then go into solar orbit, but it failed to reach the Moon instead being used to study the radiation belts.
• SCORE tested satellite communication relay broadcasting a recorded Christmas message go beyond Sputnik's beeps.

1959 would see just two Soviet orbital and deep space successes and one partial with Luna 1, 2, and 3 all with an R-7 derivative. The US had successful/partial Vanguard, Discoverer/Corona, Explorer, and Pioneer launches.

1960 would see just three Soviet orbital successes, all Vostok test flights. The US had 16 successful orbital and deep space flights.

The space race would go on like this with the US outpacing the Soviets in number of orbital launch attempts and successes, and with a larger diversity of launch vehicles.

To be clear, I'm not saying the US was "ahead". I'm rejecting the view of 1955-1970 as a linear race to put a person on the Moon measured only by cherry-picked firsts and offering a broader view. If there's a US tilt to this answer its because I'm more familiar with the US space program.